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4
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1answer
249 views

Can the Burgess-Hazen analysis of Predicative Arithmetic be extended to Transfinite Types?

Around page 300 of his book "Mathematical Thought and its Objects", Charles Parsons discusses the work of Edward Nelson, who believes that mathematical induction is impredicative, because it can be ...
23
votes
4answers
1k views

Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic

Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the ...
9
votes
1answer
681 views

Is an ultrafinitist Hilbert's program doomed?

Hilbert's program is popularly understood as an attempt to justify infinitary mathematics with a finitary consistency proof. Godel's Second Theorem is usually considered as showing this is not ...
12
votes
1answer
604 views

Can FPA really prove its consistency?

I will ask the question first and then explain. QUESTION: FPA can prove its own consistency in the Godelian sense. But can it really prove its consistency? FPA is a multi-sorted first-order ...
4
votes
3answers
474 views

Provability in Second-Order Arithmetic without the Successor Axiom

Consider second-order Peano Arithmetic Z2, i.e. the two-sorted first-order theory with induction and comprehension. Remove the assumption about the totality of the successor relationship (the ...
6
votes
4answers
711 views

Does there exist a non-trivial Ultrafinitist set theory?

Does there exist a set theory T-which has not yet been proved to be inconsistent-and in which one can prove the existence of (1) the empty set (2) sets that are singletons and (3) sets which have ...
6
votes
3answers
789 views

What is the status of irrational numbers within finitism/ultrafinitism?

According to constructivism a mathematical object to prove that it exists". There are several formulas to calculate pi, such as: so I take it ...
7
votes
1answer
510 views

Nelson natural number objects in a topos (say)

Nelson's predicative arithmetic (survey article) is a very weak system of arithmetic extending Robinson's $Q$ (Wikipedia). We can have natural number objects in a topos, or even a merely finitely ...
6
votes
1answer
868 views

How are mathematical objects defined from an ultrafinitist perspective?

I remember attending a lecture given by an ultrafinitist who denied that curves are a set of points, he would only say that any particular point may or not be on the curve. Similarly for algebraic or ...
0
votes
1answer
898 views

Is it possible to construct a finite mathematical universe? [duplicate]

Possible Duplicate: Is there any formal foundation to ultrafinitism? Very recently I have come across the skeptic opinions of a school of mathematicians(ultrafinitist) over a physically ...
2
votes
2answers
548 views

Natural numbers of great kolmogorov complexity

Before I ask my question, let me give you a mini-preamble: in 2006, during an animated discussion on feasibility, ultrafinitism, and what else on FOM, I introduced (informally, and to speak the tuth, ...
48
votes
8answers
5k views

Is there any formal foundation to ultrafinitism?

Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to wikipedia, it has been ...
50
votes
3answers
6k views

Nelson's program to show inconsistency of ZF

At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say: Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel ...