Region in hydrogen atom where an electron may be located. It has an infinite number of locations accessible to it within the probability wave lobe.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 outside to the physical world, and inquire about infinities there, or if such can even exist. In the sketch shown, we have a lobe of a probability wave (shaded) inside a Hydrogen atom. An electron of charge (-e) is shown at a Cartesian location (x,y,z) that is also referred to spherical coordinates (r, theta, phi) 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 A (angstrom) or 10^-8 cm. Then one might find changes in the measured electron’s position: 1A +/- 0.1A, 1.0001A +/- 0.0001A, 1.000002A +/- 0.00002A, 1.000003A +/- 0.000003A, 1.000004A +/- 0.00004A 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, at say 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 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, which I delivered in an earlier blog – to do with opinions and evidence- was 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 N#0 = À_o – called ‘aleph nought’ = w (actually omega but I use w since the blog interface for certain symbols is limited).
Thus: N#0 = w = oo.
Note w is an ordinal number which we identify with the set of its predecessors, N (all earlier ordinals). Now, if v and q are cardinal numbers, we calculate the cardinality v + q, by finding two sets V and Q such that V = v, Q = q, and V and Q have no elements in common, then letting v + q = V (U)Q
The interesting thing about aleph-nought and its co-transfinites (based on a proof at the end), is that it is non-denumerable if the set of operations is confined to itself. For example, if 1 were added to it:
À_o + 1 = À_o
that is, the addition does not change the total! It is still aleph-nought! Meanwhile, the product:
“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 outside to the physical world, and inquire about infinities there, or if such can even exist. In the sketch shown, we have a lobe of a probability wave (shaded) inside a Hydrogen atom. An electron of charge (-e) is shown at a Cartesian location (x,y,z) that is also referred to spherical coordinates (r, theta, phi) 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 A (angstrom) or 10^-8 cm. Then one might find changes in the measured electron’s position: 1A +/- 0.1A, 1.0001A +/- 0.0001A, 1.000002A +/- 0.00002A, 1.000003A +/- 0.000003A, 1.000004A +/- 0.00004A 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, at say 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 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, which I delivered in an earlier blog – to do with opinions and evidence- was 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 N#0 = À_o – called ‘aleph nought’ = w (actually omega but I use w since the blog interface for certain symbols is limited).
Thus: N#0 = w = oo.
Note w is an ordinal number which we identify with the set of its predecessors, N (all earlier ordinals). Now, if v and q are cardinal numbers, we calculate the cardinality v + q, by finding two sets V and Q such that V = v, Q = q, and V and Q have no elements in common, then letting v + q = V (U)Q
The interesting thing about aleph-nought and its co-transfinites (based on a proof at the end), is that it is non-denumerable if the set of operations is confined to itself. For example, if 1 were added to it:
À_o + 1 = À_o
that is, the addition does not change the total! It is still aleph-nought! Meanwhile, the product:
À_o x À_o = À_o
since a one-to-oe correspondence is exhibited between (w X w) and w.*
In addition, the transfinite number aleph-nought to the transfinite power aleph-nought:
À_o^À_o = À_1
À_o^À_o = À_1
Any non-empty set M_# is said to be "countable" if M_# < À_o
Not hard to see iff there exists a function f_wM (of w onto M_#) listing all memeber of M_#.
An exceptionally critical aspect of Cantor's transfinite cardinals putting the axe to Aquinas' statement is showing that the union of countably many countable sets is countable in his framework. Thus,
let U_nA_n mean:
A_oUA_1UA_2UA_3......
If for each n ( w, we have a set A_n and a function f_n mapping w onto A_n, then we can get a function g mapping w onto U_nA_n by letting: g(k) = f_a(b) where [a,b] is the kth pair in the listing w X w already given. Thus, g lists U_nA_n as:
{f_o(0), f_o(1), f_o(2), f_1(2), f_2(0), f_2(1), f_2(2), f_o(3).....}
This is the basis of the second number class, called II by Cantor. And the basis for the earlier statements regarded the unchanging aleph nought even when one adds aleph nought to it!
Thus, Cantor managed to define and explicate an infinite set, within which an infinity – infinite magnitudes embodied in w(omega), could be given.
* Diagrammatic proofs of all these are given in Rucker (op. cit.), pp. 249-251.
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