[FOM] Indefinitely large finite sets

[matthias]

Thanks for the valuable comments. I shortly looked at the paper about 
"Constructive nonstandard analysis without actual infinity" given by the 
link. It's more connected to Mycielski’s '81 paper and relates it to 
nonstandard analysis. I have to look at the papers mentioned by Stephen 
Simpson.

Most approaches that I have seen use an enrichment of the language with 
further constants and axioms for them. For me it is more natural to 
directly handle indefinitely large sets purely semantically without any 
manipulation of the syntax. There is no need for new dummy constants, it 
is possible that the model has a structure such that quantification runs 
over finite sets only. To this end it is enough to formally define a 
notion that a (finite) set is "sufficiently large" or "indefinitely 
large" relative to some other (finite) sets. This is the idea that I 
follow.

Stephen Simpson wrote:
<--
As regards terminology, I prefer "potential infinity" to "indefinitely 
large finite set," because it seems to me that the latter phrase is 
confusing and misleading.
-->

I should have been more precise. To me there are two concepts involved:

1) Indefinitely extensible sets. This is basically Dummett's idea that a 
reference to such a totality immediately creates a new object of it. 
Although it is often used in combination with ordinal numbers, it 
applies to natural numbers as well. Take the concept "number of 
numbers": First there is none, creating 0, then there is one number, 
creating 1 and so on. I think that Dummett also applied this notion to 
natural numbers in his late writings (and of course, he used it to argue 
for intuitionistic logic).

I think that this notion is a suitable replacement for "potential 
infinite". Lavine does not mention it in his book "understanding the 
infinite", but I think that it fits perfectly to the idea of an 
indefinitely large finite set. Formally this should be a family (M_i)_i 
of finite sets. Potential infinity sounds to me as "being beyond the 
finite", but actually it is unboundedness.

2) Indefinitely large finite sets. Provided we have an indefinitely 
extensible set (M_i)_i, this refers to some stage M_i of it. It is a 
relative notion and depends on some context. In Mycielski’s approach the 
dummy constants are used to refer to such a set.
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