[FOM] Indefinitely large finite sets

[George Cherevichenko]

For example
https://core.ac.uk/download/pdf/14916782.pdf

чт, 6 дек. 2018 г. в 03:48, matthias <matthias.eberl at mail.de>:

> 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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