New Foundations is the axiomatic set theory in Quine's 1937 article "New Foundations for Mathematical Logic"; it simplifies the theory of types of Principia Mathematica.

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Are descriptive and ontological notions of equality equal? [closed]

‎Let ‎$‎‎a$ ‎and ‎‎$‎‎b$ ‎are ‎two "‎objects". ‎What ‎is ‎the ‎meaning ‎of‎ ‎‎$a=b‎‎$‎? This is one of the deepest problems of philosophy and logic because one needs a complete information about ...
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0answers
156 views

Is there a non-trivial consistency preserving transformation?

In ‎set ‎theory ‎"equiconsistency" (and not "consistency") ‎of ‎the ‎theories ‎is the‎ ‎main ‎part ‎of ‎researches. ‎So ‎we ‎usually ‎try ‎to ‎construct a‎ ‎new model ‎using a‎ ‎given ‎one. ‎In ‎the ...
12
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3answers
773 views

Where is the end of universe?

In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is ...
8
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2answers
354 views

Definitions of ordinal besides von Neumann & Frege-Russel?

So my Google-fu didn't show any references on this. I'm studying an obscure set theory (ML, a variation on NF with proper classes) and it seems to not deal well with the standard definitions of ...
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4answers
427 views

Consistency of the concept of the collection of all collection

By Russel's paradox, we know that the concept of the set of all sets is inconsistent. Similarly, if classes have only sets as members, the concept of the class of all classes is inconsistent because ...
10
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2answers
430 views

Constructible models of New Foundations?

Hi all! Is there anything like Gödel's constructible universe for New Foundations? More precisely, I would like a process for taking a model $M$ of NF, and using it to build a model $L \subseteq M$ ...
12
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6answers
1k views

Understanding Specker's disproof of the axiom of choice in New Foundations

Hi all! I am trying to understand Specker (1953)'s proof (found here) that the axiom of choice is false in New Foundations. I am stuck on the following point. At 3.5 Specker writes: 3.5. The cardinal ...
8
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1answer
446 views

A question about Quine's set theory NF.

This question might not really be considered appropriate for mathoverflow.net but I'll risk asking it and apologize in advance if I have commited a booboo. It is often said that in NF one can prove ...
5
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1answer
335 views

In search of a set theory with specific properties

I'm in search of a set theory that satisfies the following requirements. There is a universal set $V$ such that $\forall x(x \in V)$. So for example, $V \in V$. Sets whose elements are 'large' ...
3
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0answers
326 views

New Foundations and weaker forms of choice

New Foundations (introduced by Quine) proves that $AC$ is false. Out of curiosity, is $NF$ consistent with countable choice or dependent choice? What's the strongest consequence of choice still ...
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5answers
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How much of ZFC does Quine's New Foundations prove?

Main Question: Does anyone know of a reference that can tell me which axioms of ZFC Quine's New Foundations prove, disprove, and leave undecided? Secondary Question: I've read that diagonal ...
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478 views

Limiting set theory using symmetry

[Cross-posted from here] If my understanding is correct, naive set theory needs to be restricted in order to avoid paradoxes including the Russell paradox. Typically, the restriction is expressed in ...