[FOM] Indefinitely large finite sets

[matthias]

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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