[FOM] Indefinitely large finite sets
matthias
matthias.eberl at mail.de
Wed Dec 5 02:32:36 EST 2018
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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