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Infinity in Mathematics
Fuck Infinity Part 2 by Dr. A. Crank

Since Aristotle, probably most mathematicians, have rejected the use of an actual infinity, and instead accepted the idea of a potential infinity, that one could add 1 to any natural number (to take the example of arithmetic), without reaching a last number. However, all that changes with Georg Cantor (1845-1918) and his work in transfinite set theory.

Galileo noted in 1638, that the set of all natural numbers {1, 2, 3, 4,…}, could be put in a 1-1 correspondence with one of its subsets {1, 3, 5, 7, …}, the set of odd natural numbers, even though the set of natural numbers has numbers that are not in that subset. Cantor did not see this as a paradox and said that the sets are the same size, both denumerably or countably infinite. However, by an argument known as the “diagonal argument,” he allegedly showed that the set of natural numbers was a smaller infinite set than the set of real numbers, because a parallel type of 1-1 correspondence could not be set up. In a hypothetical list, it is allegedly possible to always change a number in the diagonal of the set of reals, to create a real not in 1-1 correspondence with the natural numbers. Hence, the real numbers were of a higher cardinality than the natural numbers, nondenumerably or uncountably infinite, and that is just the start of the transfinite stairway to Platonic heaven. More on this bs below.

Paradoxes: Logical

Not all mathematicians were pleased with the arrival of the transfinite enfant terrible. The great French mathematician Henri Poincaré (1854-1912) regarded Cantor’s transfinite set theory as a “disease from which one has recovered.” But he got that one wrong.

Around the turn of the 20th century a series of logical paradoxes in set theory, such as Russell’s paradox, of the set of all sets not members of themselves (a set which is a member of itself, if and only if it is not a member of itself and which is therefore contradictory), rocked the mathematical world. Actually, just a few elites in a closed circle got “rocked;” nobody else cared a fuck. Anyway, infinity was thought to have had a part to play in this, but it was not the only problem; self-reference seemed problematic as well. Nevertheless, set theory paradoxes continued to be uncovered, such as the paradox of the set of all truths, which conflicts with one of old Cantor’s central theorems, the power set theorem. (See P. Grim, The Incomplete Universe, (1991), pp. 92-93.) Also, more below on this.

Perhaps what really did cause irritable bowel syndrome among the logicians, was that it was proven on the basis of the axioms of standard set theory that it was not possible to prove or disprove various transfinite set theoretical statements, such as whether or not there were other transfinite sets between the known ones, or even the size of certain large transfinite sets (the continuum hypothesis). It is somewhat ironic to note that one of the mathematic establishment’s leading symbolic logicians, Abraham Robinson, who himself developed a theory of infinitesimals (known as non-standard analysis), said in his 1973 retiring presidential address at the Annual Meeting of the Association for Symbolic Logic (“Metamathematical Problems,” “Journal of Symbolic Logic,” vol. 38, 1973, pp. 500-516) the following: “While others are still trying to buttress the shaky edifice of set theory, the cracks that have opened up in it have strengthened my disbelief in the reality, categoricity or objectivity, not only of set theory but also of all other infinite mathematical structures, including arithmetic.” (p. 514) I wonder if his skepticism extended to his own non-standard analysis? It should have.

Robinson said as well: “In terms of the foundation of mathematics, my position (point of view) is based on the following two main principles (or opinions); (1) No matter which semantics is applied, infinite sets do not exist (both in practice and in theory). More precisely, any description about infinite sets is simply meaningless. (2) However, we still need to conduct mathematical research as we have used to. That is, in our work, we should still treat infinite sets as if they realistically exist.” A. Robinson, “Formalism 64, Logic, Methodology, and Philosophy of Science,” in Y. Bar-Hillel (ed), “Proceedings of the 1964 International Congress,” (North Holland, 1964). But if the concept of infinite sets is really problematic, why should one therefore have confidence in their use even on pragmatic grounds? Why, therefore, should the concept of infinite sets have utility, any more than say, fairies at the bottom of the garden? Why should one trust mathematical reasoning employing the concept of an infinite set?

As an example of some of the problems that infinity raises for conventional mathematics, consider how use of the infinite raises immediate problems for classic probability theory. The probability of choosing any particular real number in the range [0,1] is zero, but some number would be chosen, so the probability, understood intuitively as the chance of a draw, is non-zero. A variant: consider a circle with a non-denumerable infinity of points. A “magic” rotating pointer is assumed to be able to stop at any point. The probability of any point being stopped at is zero, as the denominator in the probability ratio is “infinite,” yet the pointer is assumed to be able to be capable of stopping at any point. A final example. How probable is an infinite sequence of heads in an imaginary infinite number of tosses? (See “Analysis,” vol. 67, no. 3, 2007, pp. 173-180.) One argument is that the probability of heads of such a toss is ½.½. ½ …, which converges to zero. But, by another argument the set {h, h,h,…} is one possible sample space, and so is logically possible, and thus must have a non-zero probability. There are many attempts to get around these particular problems, but at this stage in the argument, the examples illustrate the problems which infinity raises.

The Great Train Wreck of the West

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PRApril 28, 2018 3:48 PM UTC

I can tell you wrote this while drinking, haha! It would help to have some links. Since you wrote this, I feel obligated to understand it.

Infinity is what divides Bayesians from classical statisticians. WM Briggs' writes in his .pdf on basic probability that classical statisticians say that, for example, 95% confidence intervals contain the mean 95% of the time given an infinite number of confidence intervals. ANother example is that distributions are said to be true given an infinite number of trials.

I guess the idea of infinity is ontologically impossible this side of the day of judgment, also. The new heavens and the new earth are going to last forever, as will the punishment of the wicked. Meanwhile, the Devil's time is short and all matter and events are countable by God. There's not an infinite number of anything countable.