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3
votes
3answers
120 views

Direct axiomatization of ordinal and cardinal numbers

Again, this question is related (**) to a previous one: in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ...
6
votes
1answer
355 views

Taller models of ZFC

This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy. Using forcing techniques, at least the ones I know of, one starts from a ...
1
vote
1answer
140 views

What restriction(s) of Goedel's primitive recursive functionals is (are) necessary and sufficient to prove the consistency of $PRA$

It is well known that one can use Goedel's primitive recursive functionals of finite type to prove the consistency of $PA$ (Peano Arithmetic). As such, one can certainly use them to prove the ...
3
votes
1answer
180 views

A question regarding the consistency of Nelson's Predicative Arithmetic

Following Dan Willard (from his paper "Self-Verifying Axiom Systems, the Incompleteness Theorem, and Related Reflection Systems", found on his homepage, pdf here): "Define an axiom system $\alpha$ ...
5
votes
1answer
289 views

Ill-founded models of set theory with well-founded ordinals

Let $(\mathcal{M},E)$ be an internally non-well-founded model of set theory i.e of $ZFC^{\neg f}=ZFC\setminus \mathrm{foundation}+\neg \mathrm{foundation}$, then $\mathcal{M}$ includes an infinite ...
1
vote
1answer
226 views

Does mathematical induction presuppose the existence of a completed infinity?

Consider the following statement by Edward Nelson--this from the "Outline" of his 'proof' of the inconsistency of $PA$ (which Terry Tao found to contain an error): "The induction axiom schema of ...
2
votes
1answer
146 views

Is there a simpler axiomatization for the quantifiers? [closed]

There is those one Q5 to Q7 in https://en.wikipedia.org/wiki/Hilbert_system#Formal_deductions But I know the axioms of Boolean algebra were simplified to this ...
6
votes
1answer
246 views

How many closed measure zero sets are needed to cover the real line, really?

This is a refinement of an earlier question. This question assumes familiarity with combinatorial cardinal characteristics of the continuum. For the reader's convenience, I reproduce below the ...
8
votes
1answer
360 views

How many closed measure zero sets are needed to cover the real line?

This question assumes familiarity with combinatorial cardinal characteristics of the continuum. Let $\mathcal{E}$ be the $\sigma$-ideal generated by closed measure zero subsets of the real line. It ...
2
votes
0answers
104 views

Some questions regarding an alteration of Grzegorczyk's theory of concatenation, $\operatorname{TC}$

Consider Grzegorczyk's concatenation theory $\operatorname{TC}$, a "weak theory of words over the two letter alphabet $\Sigma=\{a,b\}$" (this from Grzegorczyk and Zdanowski's paper Undecidability and ...
1
vote
1answer
284 views

Was “arithmetical translation” (coding in the Goedel sense) ever a part of Hilbert's Program?

Was "arithmetical translation" (that is, coding in the Goedel sense) ever a part of Hilbert's Program? I ask this question for several reasons: i) it gives the numerals |, ||, |||,.... an ersatz ...
14
votes
2answers
449 views

Can a stochastic Turing machine output a consistent extension of PA with positive probability?

Suppose that we interpret the output tape of a Turing machine as an assignment of true or false to all sentences of PA, taking the $n$th output bit as the truth value of the sentence with Goedel ...
6
votes
1answer
142 views

internalization of the concept of large and small category

I have been poking around the internet and nlab looking at the concept of large and small categories. My original focus was locally presentable categories of categories and I was thinking of finding ...
17
votes
0answers
400 views

The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?

For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
4
votes
0answers
98 views

What useful admissible rules does ZFC have beyond the deduction theorem?

I'm interested in formal proof verification, and one of the surprisingly difficult parts of this is dealing with proofs by contradiction. The issue is that the final step of such proofs is typically ...
0
votes
0answers
319 views

Set as a (strict) infinite-category?

First, let me say that I have no idea if such a post has its place here. However, I believe that the ideas I'm going to present are important. The goal of this thread is three fold: 1) trying to ...
9
votes
1answer
326 views

What is the most transparent, rigorous definition of the Univalence Axiom?

I've been studying homotopy type theory and trying to grasp the Univalence Axiom. I have yet to find a concise, accessible, rigorous definition of Univalence. I have several excellent survey papers ...
20
votes
3answers
2k views

Who needs Replacement anyway?

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in ...
5
votes
0answers
358 views

Order theory as a foundation of mathematics?

I know the followings kinds of formalization of mathematics: based on set theory (e.g. ZFC) based on type theory (e.g. the formalism of Coq proof assistant, as an advanced example) based on category ...
3
votes
0answers
102 views

Hilb as a Colimit in the Category of Scott Complete Categories (foundations)

Here is a paper I found by Adamek that generalizes Domain theory into categories of categories called Scott Complete Categories. The category of Scott Complete categories is denoted SCC. For years, ...
8
votes
0answers
219 views

Is there a notion analogous to separability but requiring definable rather than countable sets?

Among models of $\lambda$-calculus, some like the Bohm tree model have the property that every element is a directed sup of definable elements, whereas others like the $D_\infty$ and $P(\omega)$ ...
4
votes
2answers
318 views

Can one make a category concrete by “enlarging the universe”?

This is more or less a followup of this question. There, it was established that (it is well known that) the homotopy category of topological spaces is not concrete, in other words, there is no ...
-4
votes
1answer
205 views

An axiomatic system with a set of constants that form a complete ordered field [closed]

I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these ...
6
votes
1answer
682 views

Does Nelson try to prove PA inconsistent directly?

Edward Nelson is known for his serious attempts to show that Peano axioms, and sometimes even weaker theories, are inconsistent. I wasn't able to find Nelson's papers anywhere, so I wanted to ask a ...
12
votes
3answers
1k views

Proof correctness problem

I was watching this talk by Vladimir Voevodsky which was given at the Institute of Advanced Study in 2006. In his talk the first slide he shows has the following written on it: ...
5
votes
2answers
917 views

Why can't mathematics be formalised in terms of classes rather than sets? [closed]

I've always been curious about the seeming compulsion to found mathematics upon sets, be it ZF(C) or some other system. Of course, there are other approaches these days like category theory and type ...
4
votes
1answer
136 views

Class theory with support for self-application of class functions?

To every natural number $n$, we can assign its Church numeral $\underline{n}.$ A formal definition would be: $\underline{0}(f)=\mathrm{id}_{\mathrm{dom}(f)}$ $\underline{n+1}(f) = ...
1
vote
1answer
251 views

Would a non-constructible set become constructible if we had oracles of arbitrarily high cardinality for the halting problems of ordinal computers?

I still have trouble to grasp the concept of a non-constructible set, my intuition is that we could "avoid" the non-constructibility of many of them if we assume we have "ordinal computers" extended ...
8
votes
3answers
842 views

What set theoretical questions could never be answered by Turing machines of arbitrary cardinality?

Let us assume that there are Turing machines of arbitrary cardinality, by that I mean they can have input tapes of any arbitrarily high cardinality and compute for a number of steps also of ...
9
votes
3answers
859 views

Consistency of Analysis (second order arithmetic)

Is there a proof of the consistency of Analysis (second order arithmetic), which is similar to Gentzen's proof of the consistency of arithmetic? Update: Which (different) methods can be used to ...
-1
votes
1answer
422 views

What is an example of a non-axiomatic mathematical system? [closed]

In this wikipedia article on the foundations of mathematics, it says: In practice, most mathematicians ... do not work from axiomatic systems Is this correct? If so, what is an example of this?
5
votes
1answer
497 views

Why is adopting Russell's Axiom of Reducibility as strong as eliminating the Ramified Hierarchy?

In order to respond to concerns of impredicativity, Bertrand Russell developed a system of ramified second-order logic, which is like regular second-order logic except the comprehension schema is ...
1
vote
0answers
510 views

What is the role of the (formalized) omega rule in Ramified Analysis?

In the 1960's, Feferman and Schutte did groundbreaking proof-theoretic work to find out the strength of predicative systems of second-order arithmetic. They used the ramified theory of types, a ...
3
votes
2answers
312 views

What is the proof-theoretic ordinal of Hyperarithmetical Comprehension?

As I discuss in my answer here, it seems to me that Solomon Feferman shows, on pages 10-11 of his seminal 1964 paper "Systems of Predicative Analysis", that if you consider predicative second-order ...
1
vote
0answers
178 views

Has the Ramified Theory of Types been applied to Predicative Set Theories?

Questions of predicativity are well-studied in the context of arithmetic. We have a base theory, first-order Peano arithmetic. Some people, like Edward Nelson (in chapter 1 of his book) and Charles ...
5
votes
2answers
420 views

Did Gödel prove that the Ramified Theory of Types collapses at $\omega_1$?

Second-Order Arithmetic is considered impredicative, because the comprehension scheme allows formulas with bound second-order variables that range over all sets of natural numbers, including the set ...
4
votes
1answer
316 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 ...
6
votes
1answer
707 views

Original proof of Gödel's completeness theorem compared to Henkin's proof

May I have some clarification about original proof of Gödel's Completeness Theorem compared to "standard" Henkin's proof based on Model Existence Lemma ? My understanding of Gödel's original proof is ...
10
votes
2answers
454 views

Ways to define “definability”

The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : ...
2
votes
0answers
121 views

Hosting Category Theory in a “universe” that is non-LFP

WHen I did my MSc, I was trained by a very talented topologist. I had a passion for the subject before and since. Now I am interested in category theory, but I seem to be very interested in the ...
4
votes
2answers
386 views

Functor category's objects fail to be a class?

Given two categories, we can form the functor category whose objects are functors. Functors by definition consist of two mappings from in general classes to classes, which makes it fail to be a set. ...
4
votes
2answers
282 views

On wild behavior of $\omega_{1}$ in the absence of some essential axioms of $ZFC$

The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an ...
1
vote
0answers
179 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 ...
11
votes
1answer
728 views

Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?

Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense ...
12
votes
3answers
832 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 ...
2
votes
0answers
408 views

Is there a notion of “predicative given the real numbers”?

A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies ...
6
votes
3answers
418 views

Does the Feferman-Schutte analysis give a precise characterization of Predicative Second-Order Arithmetic?

A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies ...
25
votes
4answers
2k 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 ...
8
votes
0answers
562 views

Has anyone pursued Frege's idea of numbers as second-order concepts?

Gottlob Frege was a pivotal figure in the history of mathematical logic. He gave an analysis of numbers that proceeded along roughly the following lines, in his books "The Foundations of Arithmetic" ...
14
votes
4answers
678 views

Does the existence of the von Neumann hierarchy in models of Zermelo set theory with foundation imply that every set has ordinal rank?

Let $T$ be the theory consisting of Zermelo's original set theoretic axioms (extensionality, empty set, pairing, union, powerset, infinity, separation, choice) together with foundation. Put more ...