[FOM] Indefinitely large finite sets

[Joe Shipman]

The nice thing about Lavine’s interpretation is that it can be extended to give a good intuitive argument that ZFC is consistent, and a justification of the ZFC axioms, with the Axiom of Infinity replaced by an alternative “Zillion” axiom. This has the same strength as AxInf but is interpretable as a statement that there is always a large enough finite set to have desired indiscernibility properties. These can be seen as similar to the indiscernibility associated with a Russelian Reducibility axiom or a Quinean stratified comprehension axiom.

This is the best answer I know of to a question I posted here years ago, asking for an axiomatization of arithmetic or of hereditarily finite sets that was as strong as ZFC but which did not posit infinite sets. (It may not completely answer the question, because the finite sets given by Lavine’s “Zillion” axiom aren’t exactly like the HF sets, lacking certain transitivity properties.)

— JS

Sent from my iPhone

> On Dec 5, 2018, at 2:32 AM, matthias <matthias.eberl at mail.de> wrote:
> 
> I like to very shortly explain the finitistic approach of Lavine and Mycielski, since I think that it is not well known and I also think that it is interesting.
>  
> Jan Mycielski presented in a paper (JSL ’86) a finitistic interpretation of classical first order theories. He started with a finite set of formulas, usually axioms of a theory, and translated them by adding formal bounds to the quantifiers. Additionally there where axioms that bookkeep these bound. He showed that every finite consistent set of formulas, if modified in that way, has a finite model. The technical basis is as in the proof of the Löwenheim-Skolem theorem.
>  
> The intuition behind these bounds is that they increase “indefinitely”, so in formula “forall x_0 exists x_1 forall x_2 …” the x_0 refers to a finite set, say M_0, x_1 to a larger set M_1, which depends on set M_0, and x_2 to an increased set depending on M_0 and M_1. Lavine in his book “Understanding the Infinite” described this intuition in detail and applied it also to ZFC set theory.
>  
> I noticed that it is not necessary to translate formulas and add bookkeeping axioms. It is possible to directly interpret the first order formulas in a model. Basically one has to replace the infinite carrier set M by a family (M_i)_{i \in I} of finite sets and adopt the universal quantifier to run over a “sufficiently large” finite set. So this is a purely finitistic, sound and complete interpretation of a first order theory. Of course, you have to restrict the instances of an axiom scheme to a finite part, but there is no restriction in which way.
>  
> The impact seems to me that it becomes possible to understand infinite sets as indefinitely increasing finite sets rather than as completed infinite sets without any impact on the theory. Although Lavine notes that this approach is finitistic and does not need a potential infinite, I think that it is best understood when using indefinitely increasing set. The concept of an indefinitely increasing finite set is a form of finitism (I don’t like the word “potential infinite”, since it suggests that it is a kind of infinity).
> 
> The infinite is then understood as an indefinitely large part of the finite, a "relative infinite". The consequence for real numbers is this: Real numbers are no longer the same as Dedekind cuts (seen as completed infinite sets), but they can be approximated by them. Nevertheless real numbers could be introduced abstactly as objects of a complete ordered field. Regarding Tim Chow's question about introducing real numbers rigorously, it may help to see them as indefinitely increasing, but still finite Dedekind cuts.
>  
> I would also like to know whether anyone is working on these ideas or is interested in them. Any feedback is welcome.
> 
> Matthias
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