[FOM] Potential infinity
Rafal Urbaniak
rfl.urbaniak at gmail.com
Tue Mar 10 12:50:19 EDT 2015
There is some work being done on making a formal sense of potential
infinity by Marcin Mostowski and some other logicians in Warsaw. A short
paper where you can get the gist of the idea is here:
http://www.frontiersinai.com/turingfiles/May/mostowski.pdf
and some other papers are:
Michal Krynicki, Marcin Mostowski, Konrad Zdanowski:
Finite Arithmetics. Fundam. Inform. 81(1-3): 183-202 (2007)
Marcin Mostowski:
Potential Infinity and the Church Thesis. Fundam. Inform. 81(1-3): 241-248
(2007)
Marcin Mostowski
On Representing Concepts in Finite Models. Math. Log. Quart. 47 (2001) 4,
513=523
Marcin Mostowski, Konrad Zdanowski:
FM-Representability and Beyond. CiE 2005: 358-367
Marcin Mostowski, Konrad Zdanowski:
Coprimality in Finite Models. CSL 2005: 263-275
Marcin Mostowski, Anna Wasilewska:
Arithmetic of divisibility in finite models. Math. Log. Q. 50(2): 169-174
(2004)
Also, there's Marek Czarnecki's work on what happens with truth definitions
in this framework
http://students.mimuw.edu.pl/~mc208417/referaty/semantics%20in%20coprimality[en].pdf
Although, this might be not what you're looking for, because the approach
is model-theoretic.
The idea is that instead of taking the omega-sequence, one looks at its
initial segments, evaluates formulas in initial segments, and
then generally says that a formula is FM-true if there is an initial
segment such that the formula is true at that segment and all longer
segments.
The set of FM-true sentences is different than the set of arithmetical
sentences true in the standard model of arithmetic. For instance,
"there is a greatest number" comes out FM-true, while there is no
particular number n of which "n is the greatest number" is FM-true.
While it's technically interesting what happens when you play around with
this setup, I'm not sure how convincing philosophically this is. For
it seems that the quantification in the definition of FM-truth is still
quantification over an infinite set (of all initial segments).
Best regards,
Rafal Urbaniak
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