# Questions tagged [new-foundations]

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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### 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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### Can you have a type theory where there is type of all types?

Normally in a type theory, you can not have a type of all types, due to Girad's paradox. This is somewhat similar to how in set theory, you cannot have a set of all sets. Therefore, usually you just ...
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### 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 ...
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### 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 ...
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### Large cardinals arising from alternate set theories

My question is whether there are model-theoretic large cardinal axioms associated with $NFU$ - or more generally, with set theories other than $ZFC$. Large cardinal properties generally come in one ...
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### 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 $L \subseteq M$ ...
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### 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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### 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 ...
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### What drawbacks are there to using NF(U) for category theory?

In category theory, you often run into what is known as "size" issues. That is, you run into the issue that the categories you try to define are too "big" to be sets, and so you need to use classes or ...
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### 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' exist. ...
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### Anti-foundational set theory with a universal set

There are alternative set theories that allow for a universal set, e.g. NF(U), positive set theory and and topological set theory. There are also alternative set theories like ZFA that allow for the ...
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### Intersection of iterated powerset in NFU

I am interested in the existence of the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ for any given set $x$, in the context of NFU (New Foundations with Urelements). It seems to me that the ...
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### New foundation in homotopy type theory

Is there any model of NF (New Foundations) on HoTT (homotopy type theory)? Because there is a model of ZF(C) on HoTT (The HoTT book, Section 10.5) and NF on ZFC (by this Wikipedia articile), I think ...
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### NF and incompleteness

Are there any well-known statements independent of NF? And also, are there prerequisites suggesting that NF in any way, to one extent or another, are not covered by the incompleteness theorem?
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### 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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### 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 ...
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### Can NFU alone prove N to be Cantorian?

Its known that NFU + infinity proves that the set $N$ of all Frege naturals is Cantorian [see here] Question: Can the same result be proved in NFU alone?
Define recursively: $\{x\}^\emptyset = x$ $\{x\}^{n+1} = \{\{x\}^n\}$ Define: $\forall n \in N: T^n|V|= | \{ \{x\}^n : x \in V \} |$ Define: $\forall n \in N: Genuine (n) \equiv_{df} \\ \{\{x\}^n: x \... 0answers 105 views ### What known paradoxes are associated with having a type-level tuple indexed by all ordinal numbers? By type-level function it is meant a function that is definable by a stratified formula that assigns to its graph the same type it assigns to each element of its domain and to each element of its ... 0answers 36 views ### Can we have a stratified theory equivalent to NFU + Infinity + choice with atmost failure of Extensionality? It is known in NFU + Infinity + Choice, that we can partition the set$U$of all Ur-elements (empty objects other than the empty set) such that each piece is as big as the set$V$of all objects, and ... 1answer 99 views ### Can removal of extensionality avoid cardinality errors in stratified theories? Let$SF$be the schema of stratified comprehension. Take the theory$SF + Infinity + Choice + \text {Extensionality fails everywhere}$. Are the following consistent with this theory?$\forall X (|...
Is there an already known axiomtization of $NF$ that is shorter than the following axiomatic system in first order logic with equality $="$ and membership $\in"$? And what is exactly meant by ...