What are p-adic Numbers?

What are the p-adics? They're a specialized class of numbers first uncovered by Kurt Densel in the late 1800s. A key aspect is their absolute value, which depends on the prime number for which any p-adic is based. The p then means the particular prime. Given primes of 2, 3, 5, 7 - for example, one can find 2-adic, 3-adic, 5-adic and 7-adic absolute values which are always computed by taking the reciprocal of the highest multiple of p which divides into any given natural number, N.

If N has no multiples of p, then the absolute value (êp ê)  is simply 1. If we are looking at a p-adic absolute value of zero, the result is always zero. (E.g. [0]_p =ê0 ê_p = 0)

Let's look at some examples before examining more elaborate applications. Consider the 3-adic versions of: 7, 5 and 1/3. What will the 3-adic absolute values be? Compute each in turn:

[7]₃ = 1  (since there are NO multiples of 3 to form the number N = 7)

[5]₃ = 1 (for the same reason, thus: [7]₃ = [5]₃ )

[1/3]₃ = [1]₃/ [3]₃  = 1/ (1/3) = 3

since the reciprocal of 3 is 1/3 which we then divide into the numerator 1.


How about obtaining the p-adic absolute values for each of these: [4]₂  , [1/6]₂ , [1/8]₃ , and [24/25]₂

The first is pretty easy, since: [4]₂ = [2 x 2]₂ = 1/4

The next isn't too terribly difficult either:  [1/6]₂ = [1]₂/ [3 x 2]₂ =
1/(1/2) = 2


and:  [1/8]₃  = [1]₃/ [8]₃ = 1/1 = 1


Note in this case, since the denominator (8) has no 3-factors, it must be that [8]₃ = 1 .

Lastly: [24/25]₂  = [3 x 2 x 2 x 2]₂/ [25]₂ =  (1/8)/ 1 = 1/8

(Again, 25 has no multiples of 2 which can compose it, so [25]₂ = 1)

Even more intriguing are the spatial relations and differences, divergences between normal space and "p-adic" space. Consider the triangle (scalene) shown in Fig. 1 and the linear dimensions (absolute values) of its respective sides. We find: A = 4 ([4 - 0]); B = 6 ([10 - 4]); and C = 10 ([0 - 10]). Now compute the sides using 2-adic absolute values (I will assume the reader can obtain the end computations based on the previous examples):

 

for A: [0 - 4]₂ = 1/4
for B: [4 - 10]₂ = [1/4 - 1/2] = 1/4
for C: [0 - 10]₂ = [10]₂ = 1/2

Amazingly, in the p-adic context we find the counter-intuitive result that side A equals side B (= ¼). In other words, in this context, the triangle is found to be isosceles! A general rubric is that for any such computations of the p-adic absolute values contingent on a given triangle's sides - there will always be found an isosceles triangle - irrespective of how the triangle appears in normal space.

Even more bizarre results await when we examine apparently infinite series in the p-adic context. Thus, a series that first appears to go on to an infinitely large extent may be found much more different when p-adics enter. Consider the series given by the sum:

S = 1 + 5 + (5)² + (5)³ + (5)⁴ + (5)⁵ + (5)⁶ + .........

To treat S p-adically, multiply both sides by 5, then place the result under the original S and subtract:

S = 1 + 5 + (5)² + (5)³ + (5)⁴ + (5)⁵ + (5)⁶ + .........

5S = 5 +   (5)² + (5)³ + (5)⁴ + (5)⁵ + (5)⁶ + .........


Subtract term by term to get: S - 5S = 1 (all other terms above and below cancel out!)

So: -4S = 1 and S = -  ¼

In other words, the sum S is less than 1 in the p-adic venue,  totally counter-intuitive! We see that evidently the notion or concept of "closeness" emerges quite differently - certainly if we can turn an "infinite" (apparently) sum into one yielding a result less than one!

Problems for the Math Maven:

1. For the triangle in Fig. 2, use 7-adic absolute values applied to the sides of the triangle, thereby compute: A, B and C and show it is isosceles.

2. Find the value of the sum S for: 

S = 1 + 7 + (7)² + (7)³ + (7)⁴ + (7)⁵ + ....

3.  Using the series in (2) as written, "invent" a new irrational number based on the p-adic form.

Catholics Ought to Exult At Ratzinger’s Departure

Ratzinger giving a World Peace Day speech last year, condemning “moral relativism”. If he’d looked more closely at his own Church he’d actually find it in spades.

Yes, Catholics should exult at  the first abdication of a pontiff since 1415, but that’s assuming they have any sense! But we know many religionists don’t. Especially in the case of personality cult religions like Catholicism, which feature regular ‘star boys’ (i.e. pontiffs) so the tendency is for bamboozled flocks to latch onto them like goddy celebs or superstars, or even saints. Such was the case with Joseph Ratzinger’s predecessor, John Paul II – now being deemed worthy of saint-hood following many years as the Vatican’s top celeb and media star.


But instead of fawning over these very ordinary and flawed humans(adorned with merely more finery-costumes),  the flock needs to keep its eye on whether their Church is a true moral barometer, and open to change as human needs change, or it is stuck back in a medieval time warp. If the last is the case, you can be sure flimsy moral rectitude will be the norm and it will use a false moralism to try to hide its own crimes. We know, looking through history, the Church is muddied with crimes high and low

This Friday (the timing couldn’t be more perfect!) Catholics will get a look inside Ratzinger’s odious papacy with the HBO presentation of “Mea Maxima Culpa – Silence in the House of God.“   Producer Alex Gibney, on Ed Schultz last night, correctly noted Ratzinger’s resignation is his best legacy. As Gibney told Ed:

“His papacy will be haunted by the sex abuse crisis”.


On the show last night, and in the HBO documentary , Gibney documents that from 2001 – 05 Pope Rat saw every case of child sexual abuse that occurred in the U.S. – compiled in hundreds of files (including at a Wisconsin location where 200 deaf boys were sodomized by one priest). U.S. Bishops begged the Vatican to remove priests from any service or further contacts, but the Vatican refused. And where do you suppose the buck stops? With Ratzinger!

Gibney related to Ed: “What you see is patterns that emerge all over the world. Patterns are almost always of cover –ups. The Church moves forward by taking responsibility “


He added that to do this, the Vatican “needs to disgorge all documents.”

Only then, after responsibility is taken can something be done about the “Central stain on the Church”


But let’s be clear Ratzinger’s history precedes him. Church documents have revealed that as a priest Ratzinger evinced far more concern that defrocking clerics accused of sexually abusing children “could provoke some scandal among the faithful” than advocating strongly for justice for child victims. And just last year, a scandal involving leaked confidential Vatican documents portrayed the pope as a “lonely” man, plagued with trouble “keeping the shop together or getting information owing to all the filtering and intrigues surrounding him.”

Jeebus! That had to be difficult! But maybe it was his two-faced life and the strain of reconciling his moralist bromides to his Church's crimes that finally forced his abdication. At once trying to keep a lid on the sex abuse crisis which has grown like a malignancy as more and more cases have been exposed the past ten years- while portraying the outward face of moral judgment toward all sex acts of his faithful deemed as "sins". Thus, Ratzinger has had his bishops continue to crack down on Catholics of all ages simply satisfying themselves with masturbation, or sex out of wedlock –by threatening them with hellfire- while his own padres are running amuck and having their perverted way with innocents.

Let’s remind ourselves here what drives and feeds this two-faced bullshit. Recall for World Peace Day last year, Ratzinger (the former head of the Inquisition, now called “the Congregation for the Doctrine of the Faith”) lashed out at moral relativism, "the humanistic creed that holds there can be no objective standard on which to base morality". In his 2012 New Year's Day message, he warned all societies on the planet that justice and peace will remain "words without content" unless they are informed instead by the natural moral law, the key precepts of which are expressed in the Ten Commandments.

In this two -faced operation we have beheld the vile core of this Church which has thereby ceded any right to be a moral voice in the world. How?  Because it sure as shit climbs all over its people for minor sexual peccadillos (e.g. jerking off) while turning a blind eye to the major perversions occurring right under its nose, and hence endorsing them to that degree.


Let's dispense with the Vatican's time-honored appeals to "natural moral law" or "natural law" first. This is basically an antiquated relic of Catholic teaching that never seems to go away. The central problem is that it's inherently contradictory and inchoate. For example, 200 years ago invocations of "natural law" enabled the Church's moral architects to justify slavery since: "some men needed others to guide and direct them" (see e.g., Julian Pleasants in Contraception & Holiness: The Catholic Predicament, 1964, p. 88). 

Having done so, it decided to retain the sexual aspects of natural law with a rabid and unhealthy vengeance. They thereby used natural law to declare artificial contraception equivalent to "mutual masturbation" and hence meriting eternal "hellfire." Never mind that they basically placed one such act (engaged in by two loving people) — in terms of its ultimate sanction — on the same level as the vile acts of a mass murdering cannibal-rapist like Jeffrey Dahmer. Of course, this renders the whole moral foundation inchoate and absurd. You simply can't weigh the vicious acts of a serial murderer and cannibal with a loving couple using contraception on the same scales of eternal justice, no matter how brainwashed or stupid you are.

The proscription against artificial contraception is especially interesting, since Pope Paul VI’s own appointed papal commission advised against retaining it. (Op. cit.) Despite their recommendation, Paul VI rejected his commission’s advice and the Church remained steeped in its medieval past, even as the world population has more than double since then, see also: http://brane-space.blogspot.com/2011/09/human-population-growth-is.html

But this is why the Church’s current sexual morality is false (see a far more authentic sexual morality in Sister Margaret Farley’s book, ‘Just Love’). It’s false, first because the paramount objective is not enticing the RC faithful to "do good" but to control their gonads! Any institution that can control people's sex organs, controls their lives, thinking and most fundamental decisions.  Second, it's false because it ignores all the critical indicators which disclose our limited planetary resources cannot support a pseudo-ethic which encourages reproduction as the final object of any sex act. According to the site Global Footpoint Network, at:

http://www.footprintnetwork.org/

we currently need not one but one and one half  EARTHS to sustain our current rate of consumption. This means it requires on average 1.5 years for the Earth to regenerate the resources humanity uses in one year. Clearly, at its rate of extrapolation, we'd need three Earth's by 2050 - which simply isn't going to occur short of colonizing another two earthlike planets.

Even now, most people don't comprehend or process that adding 2.5 billion humans by 2050 is the carbon equivalent of adding two United States to the Earth. The total added population mass with its CO2 pollution, incepted loss of habitat and drain on water supplies, will be an unmitigated disaster. But instead of facing reality, people's brains - most - are wired for what I call "technological Pollyannism" - hoping and pining for some techno-fix that will make everything all right at the last moment. Well, don't hold your collective breath, kiddies!

But what is the Catholic Church’s response?  "No, NO and NO! We don't care how few resources there are left on Earth or how many people! You still can’t practice artificial contraception"!  They will allow you to use a pseudo-form called “rhythm method” that requires precise temperature taking of the female for critical days in her cycle! In other words, the Church is in favor of overpopulation, famine, starvation and mass depredation ultimately leading to population crash and "die off", e.g.

http://www.dieoff.org/  

With Ratzinger gone, it would be nice to think the new pontiff will be a man of change and move the Church forward into the 21st century. But don’t hold your breath. All the guys that are most likely to be picked in the upcoming Vatican Conclave are mainly members of an inner circle hand-picked by Ratzinger, so it’s doubtful they will reflect any different thinking – only more false natural law moralism.

The last REAL potential for change arrived with the selection of Albino Luciani as Pope Paul I back in 1978. He even planned to remove the binding proscription on artificial birth control. Trouble was, he also planned to investigate the goings on at the Vatican Bank under Paul Marcinkus and his contacts with Licio Gelli. Alas, he didn’t last a month, and the scoop is he was assassinated by the same sort of Vatican insiders that have strong armed Vatican conservatism on these moral issues the past three –plus decades. For more on the background to the hit on Pope Paul I, see David Yallop’s ‘In God’s Name’ .  Of course, the Vatican has religiously denied all of Yallop’s impeccably documented claims, but they’ve never disproved them. But maybe, as occurred some 35 years ago, a true reformer will somehow get in, and this time live to tell about it as well as redeem this utterly lost Church.

Asteroid Hit or Solar Meridian-Centered CME? I’d Take the Asteroid!

The CME captured by the Solar Dynamics and Heliospheric Observatory (from spaceweather.com)

As the asteroid 2012 DA14 approaches Earth, and is set to pass within 17,000 miles (or less than two Earth diameters) on Friday, at a speed of 17,400 mph, a high probability has also arisen of being struck by a CME (coronal mass ejection) three days earlier. This elicits the question- assuming the worst in each case- of which you’d rather have: a strike by a 15 story wide, 130,000 ton asteroid, or being struck by a CME.

First some background: In the case of the asteroid, NASA reports that this is the closest known approach of an object of this size. (They also add there’s no chance whatever of the thing whacking Earth. Well, assuming it doesn’t get deflected after hitting a geostationary satellite). It was discovered in February 2012 by astronomers at the Observatorio Astronómico de La Sagra; it passed the Earth on Feb. 16th of that year, missing us by a comfortable 2.6 million km (1.6 million miles). The asteroid has a very Earth-like orbit, talking 366 days to orbit the Sun. Its orbit is slightly more elliptical than ours and tilted by about 11° to ours as well. All that’s about to change though: After the encounter this year, the Earth’s gravity will change the orbit of DA14 quite a bit, reducing the period to about 317 days.

The asteroid, with a mass of possibly 10 ⁹  kg would – if it impacts - be about equal to a 20 megaton thermonuclear weapon blast. It obviously would cause considerable damage if it smacked into a city or an island like Barbados – but those odds are even less than our being struck in the first place. The odds are also 3 in 4 that if it did impact it would be an ocean strike, possibly incepting a 20-40 m tidal wave depending on the location, angle of entry, etc.

The CME, meanwhile, was captured by the Solar and Heliospheric Observatory (see graphic)  as it shot away from the Sun at 800 km/s. The initial prediction is that the bulk of the ejected mass will sail over the Earth’s north pole. However, spaceweather.com warns there is still a chance of a glancing blow. The former likely means high latitude observers on Earth will see brilliant auroras, but will experience no serious disturbance. A glancing blow possibly means one or more power grids knocked out such as occurred in Quebec in 1989 after a giant solar flare.

In a 2010 episode (‘Magnetic Storms’) of the series ‘The Universe’, the initial sequence featured a March, 2013 CME event which also triggered brilliant auroras, as far south as Cuba, but which also within hours delivered a head on blow to Earth, that knocked out the entire power grid of the U.S. and every single transformer. Such possibilities were discussed in an earlier blog, e.g. http://www.brane-space.blogspot.com/2012/06/cme-that-you-dont-want-to-see.html


The result of the CME hit depicted then was no power across the U.S. No water, since water in your taps requires electricity to pump it, and no fuel either or gasoline, since gas stations also require electricity to effect pumping. For what awaits us, get another gander at those images after superstorm Sandy Struck NY and NJ in November. Long lines, squabbles over position in lines, fights, and even supermarkets going out of business. In other words, no food either! Such a CME would also knock out GPS satellites, and that would wreak even more havoc. Computer networks everywhere would go down, and without electricity anyway – computers would become museum pieces.

The episode projected some 10-15 million dead since the power wouldn’t be able to come back for years, given all the infrastructure would need to be rebuilt, including transformers, power stations, etc. Imagine then a nation with no power to speak of, hundreds of millions without food, limited canned stores – if they have that, no water, and no way to access critical emergency networks or services. The most vulnerable, as in the case of Sandy, would be the elderly, and sick.


The episode went on to further conjecture of an even worse scenario:  a CME striking at exactly the time when the Earth’s magnetic poles are reversing. With such a reverse there’d be days or weeks with no magnetic field to protect us from the full brunt of high energy solar particles so the entire planet would be affected with up to 2-3 billion dead as a result of all the power knockouts, the starvation to follow, the mayhem. The ‘End of the World’? Technically no, but many would think it was.

So, on that basis, and given my druthers, I’d always pick the hit from an asteroid like 2012 DA14 rather than a full smackdown from a CME.

Let’s hope neither of them transpires!

A Test on Evolution for Creationists and Hard Core Fundies

The Fundies and Creationists, most of them, have never taken any courses in evolution - and yet feel it is their wont to yap all they want about it, ignorantly criticizing and commenting on what they clearly don't understand. (Heck, most haven't even read Darwin's Origin of Species)


In the interest of intellectual honesty, therefore, I invite these fundie commentators to take the following test to ascertain if they are really qualified to talk about evolution, far less criticize it.

I regard a qualification to talk (write) about the subject (say on their blogs) getting at least 7 of 10 correct. I regard a qualification to criticize it, getting at least 8 1/2 of 10 correct.

----Start test (Time - 1 hour)

1) A fossilized animal bone specimen is found to contain 0.125 gram of an unnamed radioactive isotope. Analysis reveals that the original mass was 1 gram. What is the half-life of the sample if it is dated at 3 million years?


2)Which of these would be involved in molecular evolution?

A) tRNA

 B)mRNA

 C) rRNA

D) All the preceding


3) The mutation rate for human albinism is u = 0.00001. (e.g. 1 in every 100,000). If the equilibrium frequency q = 0.01, what is the reduction in fitness?


4) Blatella Germanica, the German cockroach has the following identified alleles:

D- resistance to the pesticide dieldrin

d- non-resistance

After some defined time, the population exhibits the following three genotypes:

DD, Dd and dd

If over time, each dd and Dd roach produces one offspring, and each DD produces TWO, then find:

a) the relative fitness for the DD roaches

b) the relative fitness for the Dd roaches

c) the relative fitness for the dd roaches


5) Given the same information as above, find:

a) the selection value for the DD roaches

b) the selection value for the Dd roaches

c) the selection value for the dd roaches


6. a) Based on your analysis from 4, 5 which allele of the German roach is “deleterious”?

b) Over successive generations of German roaches it can be shown that the gene frequency of the deleterious allele will be decreased by:

-Dq = - spq / (1 – sq)²

where p is the gene frequency for the favored allele ( = 0.60)

If it reproduces at a ratio of 3:2 relative to the disadvantaged allele., use this information to find:

The gene frequency (q) of the disadvantaged allele for the roach, and thence, the magnitude of -Dq.


7) For the same example, above, estimate the values of p and q when

-Dq = -0.07

State any assumptions made.


8) In DNA there are always FOUR bases: A (adenine), C (cytosine), T (thymine) and G (guanine)

WHICH one of the following complementary base pairings would NEVER occur?

A) A-T
B) G-C

C) C-T

D) NONE of the preceding


9) Micro-evolution is allegedly tested on the bacterium E. Coli, grown in a medium devoid of galactosides, so that its three available proteins are synthesized at an exceedingly slow rate – about 1 molecule every five generations.

If galactoside is now introduced into the medium the rate of synthesis of all three proteins is increased a thousandfold. If it is withdrawn, the synthesis rate returns to its original low form.

Which of the following components is critical to the process described above?

A) a “regulator” gene (i)

B) a repressor protein R

C) An operator segment of the DNA

D) a molecule of inducer galactoside b

E) All the preceding


10. Explain why the preceding example is not really an illustration of microevolution. Define micro-evolution and provide the testable criteria for a genuine example of your choice. Be sure to include in the example how it shows microevolution and not macro-evolution

-------------------------------
Time Limit is 45 minutes (One hour with reading time included to ensure one understands what's asked)  and obviously, NO Googling! Score at least 85% and you are qualified to critique evolution – otherwise, stick to the Bible!

PUH-LEEZE Tell Me People Can't be THIS Stupid!

Al Gore is a charlatan, and “O’Bummer” just plain lies and wants to take your money. So writes a Reich wing nut on one of his latest blogs where he takes up global warming and proclaims: “There AIN’T NONE – since we can see a BLIZZARD breaking out now!” Then referring to the blizzard dumping up to 36” of snow in some places, including CT, RI and MA.

First clue, Sherlock: blizzards are SUPPOSED to occur in the winter! Last I checked, the calendar still said FEBRUARY – not July, so your ridiculous claims that Gore is a charlatan or Obama’s climate change remarks (at his Inauguration) are seriously misplaced. Maybe where the Sun doesn’t shine?

Second, singular events like this blizzard in the NE and 'Sandy' back in November, do not in general denote evidence for global warming-climate change or lack of evidence, although the increase in storm intensity over the past three decades does suggest it. The point is that in determining global warming –climate change we don’t merely look at one or two odd statistical anomalies but overall trends. Those trends as we see them are for higher and higher temperatures globally, not lower.

Our confused blogger perhaps missed the news from two months ago (at the DOHA Conference on Climate Change in Qatar), that nearly all the attendees agreed that the original goal of halting warming to a 2 degree increase is essentially fruitless. As I noted in an earlier blog: http://brane-space.blogspot.com/2012/11/bill-mckibbens-prediction-of-nightmare.html

this level is the tipping point beyond which we're all for the high jump. But the consensus now is that the international goal cannot be reached. Last year alone, the combined nations of the planet pumped out nearly 38.2 billion tons of CO2 into the air from the burning of fossil fuels such as coal and oil. This is according to new calculations published in Climate Science journal.

This amount is more than the previous year and translates into 2.4 million lbs of CO2 released into the atmosphere each second.

The cumulative effect is even more dire, given most CO2 remains in the atmosphere for at least 100 years. Thus, the CO2 now heating our world via the greenhouse effect has been accumulated since at least 1912. Given this time burden, and what's still to unfold, it's preposterous to believe temperature increases can be held to less than 2C. This is according to Glen Peters of the Center for International Climate and Environmental Research in Oslo, Norway.Andrew Weaver, a climate scientist at the University of Victoria, Canada, put it in even starker terms:

"We are losing our ability to get a handle on the global warming problem."

Translation: we are on the cusp of a massive climate catastrophe for which humans will have no solutions and for which all of humanity will pay a steep price in the coming decades....until the runaway Greenhouse turns the planet into another Venus, see e.g. http://brane-space.blogspot.com/2012/08/how-did-once-good-planet-go-bad.html


All of these dire reports, such as the 2C limits exposed at DOHA, were confirmed in the recent issue of Physics Today (p. 17) noting the trepidation with which climate modelers are viewing the rapid melting of the polar ice. See also: http://brane-space.blogspot.com/2012/12/w-antarcrtic-ice-sheet-melting-at-twice.html
The Physics Today report asserted that scientists’ alarm is justifiably based on the fact that the Arctic is the “thermostat” of the planet. Once that ceases to function, we are basically open to not just wild variations of climate (such as extreme blizzards in the winter, and extreme drought, or downpours-hurricanes and heat waves in the summer) but a progressive heating that will not only melt both polar ice caps adding meters to the ocean depths but also possible runaway heating.

The blogger may also have missed the news, or maybe he’s educationally unable to process it that 2013 had barely started but the photo image shows in an earlier blog http://brane-space.blogspot.com/2013/01/a-vision-of-humanitys-near-future.html

showed what we may likely face in the Northern hemisphere come June, July. For the record (no pun intended) July 2012 was the hottest ever recorded in U.S. annals with a mean tempeature of 77.6F. The average annual U.S. temperature for 2012 was 55.32F or one full degree higher than the old record set in 1998. (So much for the codswallop that all temperature increases ceased in 1998. But as I showed in earlier blogs, 2011-12 and onward would bring even higher temperatures if all the graphs presented in a 1998 Nature paper were properly interpreted.)

Even more eye-popping, the year 2012 displayed an average temperature a full 3.2 degrees hotter than the average for the entire 20th century! Meanwhile, nineteen states set yearly heat records - though Alaska was a bit cooler than average.

According to the National Climate Data Center (whose completed data comes out this coming week) through the first 11 months of 2012 the world was on pace to have its eighth warmest year ever. More disturbing still, 16 consecutive months have seen the U.S. with higher than average temperatures. This year promises to be no different, and we can expect not only severe heat waves such as transpired last summer and which erupted in Australia last month, but many more wildfires here in the Rocky Mountain West, as well as more intense hurricanes or superstorms like Sandy.

Instead of mocking global warming on account of  blizzards occurring in February (which  is a seasonal month for them to naturally occur in the northeast - even big ones) and making fun of Al Gore, the Reichie blogger would do better do open up a few actual physics books – because then maybe….MAYBE…he will understand climate change better, instead of making a moron of himself.

Better yet - maybe he can show he's qualified to even comment on physics issues (e.g. Big Bang, dark matter-energy, climate change) by taking and passing the physics test shown here:

http://www.brane-space.blogspot.com/2010/10/basic-physics-test.html

The time limit is one hour and thirty minutes and obviously, no googling! Score less than 70% and better stick to comic books or Bibles!

How to Count to Infinity (oo)

The electron in a hydrogen atom exhibits positions which illustrate a physical infinity.

It is a curious feature of the human brain that concepts of “infinite” and “infinity” arose almost from the instant numbers were conceptualized – and then dramatically reinforced as the printed word emerged via the Gutenberg printing press. Of course, “infinite” measures were likely originally used first in the realm of theologians. For example, Thomas Aquinas is reputed to have written (cf. ‘Infinity and the Mind’, by Rudy Rucker, p. 52):

“The existence of an actually infinite multitude is impossible”

He never framed his basis for this statement in terms of a mathematical argument, however, one supposes as does Rucker (ibid.) that what he meant is that no infinite numbers would be able to exist if there were no infinite sets. Since he could not conceive of infinite sets, then it follows there could be no infinite numbers.

Before returning to the math, let’s venture into the physical world, and inquire about infinities there, or if such can even exist. In the sketch shown above, we have a lobe of a probability wave (shaded) inside a Hydrogen atom. An electron of charge (-e) is shown at Cartesian location (x,y,z) that is also referred to spherical coordinates (r, q, f ) since we are configuring the H-atom in spherical shape.   The lobe defines the space of the atom within which the electron may appear or be located at any instant of time, t. Technically, the electron, according to modern quantum theory (not the old Bohr solar system model) can be found anywhere within the lobe volume.  

Now, while it is true that the lobe occupies a finite volume inside the hydrogen atom, it is also true that the electron can occupy an infinite number of positions within it. For example let the position at r be denoted by 1 Å (angstrom) or 10^-8 cm. Then one might find changes in the measured electron’s position defined such that:   1 Å + 0.1 Å, 1.0001 Å + 0.0001 Å, 1.000002 Å + 0.00002 Å, 1.000003 Å + 0.000003 Å, 1.000004 Å + 0.00004 Å   and so on, ad infinitum.

In other words, an infinite series. This is not mysterious and one of the first paradoxes discovered (compliments of Zeno) is that a finite length can embody an infinite series. Consider the length between 0 and 1 shown, and the units could be whatever we want, whether mm, cm, m or whatnot.

0 )-----!-x---!---------------------------------------( 1

Now, consider the point x= 0.12.It may be said to correspond to the point equal to 1/10 + 2/100 in the second sub-interval of the line defined by sets of 0.01 each. At the same time it can be said to be in the first subinterval where the sets are defined as 0.10 in size each. (Again the units are neither here nor there). For convenience and to better fix ideas I have shown the bound points of 0.1 and 0.2 for the “tenths” subintervals.

For any such decimal fraction (lying between 0 and 1) one may then use the representative form:

f = Z + a₁(0.1) + a₂(0.01) + a₃(0.001) + a₄ (0.0001)

+………a_n(10^-n)

where Z is a starting integer, or zero. (For example, in our case, Z = 0 but in another case it might be 4, and we could look between 4 and 5). The a’s meanwhile are numbers representing tenths, hundredths, thousandths, and so on. For example, a₃ – from 1 to 9 – assigns numbers in the thousandths. The point is that there are an infinite number of subintervals which could be assigned between 0 and 1. For example, between:

0.00000000000010 and 0.000000000000101,

and so on…..

Now imagine an electron between 0 and 1, with possible measurements that might range from:

0.2 + 0.1 to 0.25 + 0.01…..0.235 + 0.001…..0.2356 + 0.0001…..to 0.235608497 + 000000001

And so on and so forth. In other words though an infinite precision is impossible, an electron can technically be said to occupy ANY of the infinitely many points between 0 and 1 whose distance from zero is a terminating decimal.

Thus does the electron show the plausibility of a physical infinity. (Another example is the infinite number of wave functions that could be correlated to an electron’s position moving from an electron gun to a diffraction screen. )

Back to Aquinas’ claim of impossibility for an actual physical multitude. This was basically demolished by George Cantor’s transfinite cardinal numbers. The key move by Cantor was in making his cardinal number À_o –  called ‘aleph nought’ = W.

Thus: À_o = W = ¥

Note W is an ordinal number which we identify with the set of its predecessors, N (all earlier ordinals).  Now, if k and q are cardinal numbers, we calculate the cardinality k + q, by finding two sets K and Q such that K = k, Q = q, and K and Q have no elements in common, then letting:  k + q = K ( È)Q

The interesting thing about aleph-nought and its co-transfinites, is that it is non-denumerable if the set of operations is confined to itself. For example, if 1 were added to À_o:

À_o + 1 = À_o

that is, the addition does not change the total! It is still aleph-nought!  In addition, the transfinite number aleph-nought to the transfinite power aleph-nought:

À_o^Ào  = À₁

Thus, Cantor managed to define and explicate an infinite set, within which an infinity – infinite magnitudes embodied in W  could be given.

Universal Salvation Belief a One Way Trip to “Hell”? HELL NO!

Sign from a Fundie's blog warning any who accept universal salvation to watch out! Their final destination is HELL!

According to Wikipedia, Universal Reconciliation or Salvation is:


“the doctrine that all sinful and alienated human souls—because of divine love and mercy—will ultimately be reconciled to God.

Universal salvation may be related to the perception of a problem of Hell,, standing opposed to ideas such as endless conscious torment in Hell, but may also include a period of finite punishment similar to a state of purgatory.Believers in universal reconciliation may support the view that while there may be a real "Hell" of some kind, it is neither a place of endless suffering nor a place where the spirits of human beings are ultimately 'annihilated' after enduring the just amount of divine retribution”

I first became acquainted with universal salvation while debating an Anglican Priest in Barbados in 1982. At one point I mocked the concept of “Hell” as infantile, and basically “apartheid theology” based on gloating over the suffering of those unlike you, who don’t subscribe to your special beliefs- but whom you condemn to everlasting perdition. I also mocked the notion that the presumption of Hell meant every sect or Christian denomination could, in effect, pass judgment on every other sect or denomination. Fundies could hell bind Catholics, Catholics could hell bind Fundies, both could also consign Jews and Muslims to the everlasting pits as the Muslims could return the favor to Christians! It was an afterlife MUTUAL ANNIHILATION PACT!


The priest laughed and conceded the point, and I won the debate – scored by objective observers – based on that line of attack. After the debate, however, the priest confided to me that in the upper echelons of the Anglican hierarchy (recall here that King James, of King James Bible fame was an Anglican too) no one took Hell seriously and there was no doctrine of Hell or of exceptional salvation for the few. Instead all Anglican seminaries taught Universal Reconciliation – Salvation. The priest told me he didn’t teach it himself nor did any other Anglican padres he knew. They had to instead teach Heaven or Hell because that is what their parishioners expected. He told me that if they attempted to teach the actual doctrine of universal salvation most of the faithful would leave that Church because their skewed sense of moral right and wrong dictated the “evil” receive the ultimate punishment. But he said this dichotomy troubled him no end!


Well, no wonder! It had to be mentally pathological to know what your religion’s actual doctrine is on the one hand but not be able to openly teach it on the other. Schizoid personality disorder anyone?

But he has a point. Because most believers function at the level of children, and need a cosmic daddy to “take the rod” to the bad boys or girls, they cannot accept the conceptual notion of an afterlife wherein everyone is brought together. Even many Christian leaders who ought to know better buy into this codswallop. In a memorable final Phil Donahue show on MSNBC in 2002(still have it on videotape), he included people from five major religions, including the head of the Southern Baptists, and a leading Rabbi. When Donahue brought up the thorny issue (which most evangelicals prefer to sweep under the rug) of what happens to the Jews when ”Armageddon” is all over and the "Antichrist" (sic) is defeated, the Southern Baptist fundie didn’t mince words:

“Most will be bound for the fires of Hell, because it is appointed that only 144,000  Jews shall be saved. “

Donahue’s eyes rolled twice over then nearly popped out, as did the Rabbi’s who then entered into a furious exchange with this bigot, prefaced by the words:

“How DARE you! How dare you decide to dispatch Jews into afterlife fires when over six million have already been roasted in Hitler’s crematoria!”

The Southern Baptist didn’t blink, merely responding mechanically:

“I don’t decide that. It’s the word of the bible, not mine. Unless a man be born again and believe on the Lord Jesus Christ he cannot enter the gates of Heaven. Since nearly all Jews reject Jesus as Savior, they can’t enter the heavenly kingdom”.

This pompous display of assholinity prompted Phil D. to quip:

“Yeah, but it’s all fine and dandy that Israel be used as the setting for Armageddon in the Valley of Megiddo. Once Israel fulfills its biblical prophecies, then I guess it’s perfectly okay to send all the Jews to Hell…..oh, except for the special 144,000. And by the way, what’s so special about them? Why do they get a heavenly pass and not the others?”


The Southern Baptist  (Albert Mohler) was speechless. Or rather, perhaps so dumbfounded he wasn’t able to respond. Just as well, as the Rabbi was about to lacerate him verbally!

Now, just so we are clear, as an atheist I lean toward the probability that when you’re dead, you’re dead. I am not dogmatic or absolutist about it, but that's my take. (And no, I've seen no convincing evidence for 'near death experiences' which in any case, have nada to do with actual death!) That’s it! Finito! However, in previous blogs I have left a minuscule probability that human consciousness can survive death but in a nonlocal form, embodied in quantum wave energy. See, e.g. http://brane-space.blogspot.com/2012/07/has-stuart-hameroff-explicated-way-out.html

However, for the sake of this blog, let’s argue from the believers’ perspective that there is a conscious personal afterlife and a putative deity to preside over it. We assume the form of this deity is in line with the attribute most often declared by deity worshippers, i.e. "infinite".

IF God as infinite is taken literally, this can only mean there is no place where he isn't. Either he is infinite or not infinite. If he is infinite, and HELL also exists, then Hell must be part of the same infinity. It cannot be isolated from it or else we have a condition where Hell is apart from God's being. If this be so, then there exists a domain ("Hell") where God isn't, so he can't be infinite! In other words, God is actually a limited being.

Another way to put this is that something (Hell)  exists in addition to God's being  hence that being must be de facto delimited. If delimited it can't be infinite.

So, we have these ineffable logical conclusions:

1) EITHER - Hell exists but GOD is not infinite.

OR

2) GOD is infinite, and Hell must exist WITHIN God. (But Heaven does also)


Up to now, no fundie or evangelical  has been clever enough to resolve the Hell-God=infinite paradox. There could be two reasons: 1) none of them is smart enough to do it, or 2) they know deep down in their little atavistic pea -brains that the paradox is logically insoluble because their assumptions are in error. (Or, just as bad, they allow the blatherings in an ancient book to trump the importance of examining the assumptions.)

Thus, if they want to preserve their "Hell" they will have to admit their God isn't infinite. If they demand God to be infinite, they will have to jettison their "Hell'.

Beyond this, if modern quantum mechanics is taken seriously, then reality is one universal whole (called “nonlocal” in QM terminology) and there are no separates or localities – say wherein something can occur that doesn’t affect the rest of reality. Hence, concepts such as David Bohm’s Holomovement come front and center and these impersonal concepts can supersede that of an angry old white man in the skies that hurls unbelievers and do-badders into perdition.

In effect, it appears those sects and denominations that accept the Universalist perspective are more likely to be au fait with at least the basic underlying theme of the wholeness and unity of reality, as demonstrated for example in the Aspect experiment (1982). That set up is depicted below:

A1 ( ) <--------->( )A2


The idea being that twin photons appear interconnected (via their polarization detection) at the analyzers A1 and A2. Some have misconstrued this as violating Einsteinian relativity (e.g. fast than light signal propagation) but the superior interpretation by physicist David Bohm holds that the two photons were always connected in a higher dimensional reality, which he has called the “implicate order”.

If there is an implicate order, which is also holographic, then it very well means the Universalists have been right all along ….assuming there is a God inhabiting this implicate order, of course! Maybe the fundies and their ilk need to take more seriously the words of Philosopher Alan Watts ('Behold the Spirit', p. 244):

"Taken literally as a doctrine of everlasting torture, 'Hell' amounts to blasphemy itself- for if it were in the power of human freedom-choice to produce a consequence of this kind there must be something radically diabolical in the order created by God, and in God himself. Such an interpretation may vindicate human freedom but at the cost of demonizing God."