**2**

votes

**0**answers

101 views

### Can the Kunen inconsistency (or the existence of Reinhardt cardinals) be 'properly formulated' in Ackermann set theory?

In their paper "Generalizations of the Kunen Inconsistency" (arXiv:1106.1951v1 [math.LO]10 Jun. 2011), Hamkins, Kirmayer, and Perlmutter write the following:
The first [metamathematical issue--my ...

**-2**

votes

**1**answer

188 views

### Critical points and the Foundation Axiom

(Note: This question is related to my previous mathoverflow question, "Critical Points in $ZF$ without Choice".)
In the Stanford Encyclopedia of Philosophy entry "Non-Wellfounded Set Theory" (...

**2**

votes

**2**answers

944 views

### Unreasonable application of mathematics to the other areas [closed]

What are some papers or talks on the philosophy of mathematics which contains some statements about the unnecessary and unreasonable application of mathematics in other areas of science?
I found ...

**33**

votes

**4**answers

4k views

### Did Euler prove theorems by example?

In his 2014 book, Giovanni Ferraro writes at beginning of chapter 1, section 1 on page 7:
Capitolo I
Esempi e metodi dimostrativi
Introduzione
In The Calculus as Algebraic ...

**1**

vote

**0**answers

146 views

### Can Dedekind's 'proof' of the existence of infinite sets be properly formulated and carried out in positive set theory?

This question is related to Mikhail Katz's recent mathoverflow question, "Has Dedekind's proof of the existence of existence of infinite sets been analyzed by historians?". Dedekind's 'proof' seems (...

**1**

vote

**0**answers

98 views

### Can Basic Law $V$ be derived from Leibniz's Law in Second-Order Logic without comprehension principles?

Consider Basic Law $V$:
$\hat x$$F$($x$)=$\hat x$$G$($x$)$\equiv$($\forall$$x$)($F$$x$$\equiv$$G$$x$)
At first glance, it seems to have the same form as Leibniz's law
$x$=$y$$\equiv$($\forall$$F$)($...

**11**

votes

**3**answers

2k views

### Has Dedekind's proof of existence of infinite sets been analyzed by historians?

This pdf by David Joyce notes that in paragraph 66 of his famous essay, Dedekind claims to prove the existence of an infinite set.
The proof exploits the assumption that there exists a set $S$ of all ...

**31**

votes

**2**answers

4k views

### Why is this new result such a big deal?

This popular article reports a recent result in reverse mathematics, showing that a certain theorem in Ramsey theory is provable from RCA$_0$, the base theory in SOSOA. Then there are a bunch of ...

**2**

votes

**2**answers

371 views

### What are the sense and reference of the propositions $R \notin R$, $R \in R$, where $R=\{x \mid x \notin x\}$ in Frege's Grundgesetze?

In the paper,
Aldo Antonelli and Robert May, Frege's new science, Notre Dame J. Form. Log. 41 (2000), no. 3, 242–-270, MR 1943495.
the authors give the following quote of Frege, from his paper "&...

**22**

votes

**3**answers

1k views

### Mathematicians with Aphantasia (Inability to Visualize Things in One's Mind)

Are there any mathematicians with aphantasia? If so, could they please elaborate upon what their experience with mathematics is like?
I realize that this question probably falls outside of the scope ...

**1**

vote

**1**answer

218 views

### Have some works by Émile Borel ever been translated from French to English or another foreign language?

I plan to submit a couple of questions around Émile Borel's works in probability theory to MO.
In this scope, I'd like to know if the following works have ever been translated from French to English ...

**5**

votes

**1**answer

210 views

### Identity types: What makes Intuitionistic Type Theory *intuitionistic*?

In the opening passage of Martin-Löf's (1975) he famously says that
"the theory of types with which we shall be concerned is intended to be a full scale system for formalizing intuitionistic ...

**14**

votes

**1**answer

492 views

### Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?

Background. The field of Borel equivalence relation theory
provides a robust, unifying theory that organizes most of the
classification problems of classical mathematics into a hierarchy,
allowing us ...

**6**

votes

**1**answer

442 views

### Are there 'finitistic' nonrecursive functions (assuming Church's Thesis is false)?

[Note: In what follows, I will be using the same type of argument Laszlo Kalmar did in his paper "An Argument Against the Plausibility of Church's Thesis" found in Constructivity in Mathematics, (...

**2**

votes

**1**answer

519 views

### Can Turing machines clarify mathematical, philosophical, and physical existence?

From Harvey Friedman's manuscript on "Order Invariant Relations and Incompleteness":
DEFINITION 4.4. A $\Pi_1^0$ sentence is a sentence asserting that some given Turing machine never halts at the ...

**10**

votes

**3**answers

577 views

### The universe of sets, existential quantification in set theory

Yesterday, I posted a question that was received in a different way than I intended it. I would like to ask it again by adding some context.
In ZF one can prove $\not\exists x (\forall y (y\in x)).$ ...

**1**

vote

**1**answer

178 views

### In what sense is the “descending chain principle” for ordinals less than $\epsilon_0$ 'infinitary?

In the introduction to his paper "Assignment of Ordinals to Terms for Primitive Recursive Functionals of Finite Type", W.A. Howard writes:
Gentzen...showed that the consistency of first order (...

**2**

votes

**0**answers

211 views

### The theory of frames and locales as elementary topology [closed]

In What is elementary geometry? (pdf) Alfred Tarski defined elementary geometry to be
that part of Euclidean geometry which can be formulated and established without the help of any set-...

**4**

votes

**0**answers

307 views

### About the “semi-classical” view of Prof. Weaver and Prof. Feferman [closed]

In the thread "Is platonism regarding arithmetic consistent with the multiverse view in set theory?", Prof. Hamkins writes:
The view you are suggesting is something close to what is held by ...

**7**

votes

**1**answer

449 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 ...

**8**

votes

**2**answers

772 views

### Is platonism regarding arithmetic consistent with the multiverse view in set theory?

A "truth" platonist for arithmetic believes, given a statement in the language of arithmetic, that the problem whether the statement is true has a definite answer.
Prof. Hamkins has argued for a ...

**1**

vote

**1**answer

161 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 ...

**5**

votes

**3**answers

713 views

### What does the axiom of replacement mean and why should I believe it?

Here Professor Blass describes the following cumulative hierarchy of sets:
Begin with some non-set entities called atoms ("some" could be "none" if you want a world consisting exclusively of sets),...

**2**

votes

**1**answer

1k views

### What is the modern consensus on the difficulty of infinitesimals?

At a related thread at MSE an expert in reverse mathematics noted that "As the modern consensus is that only nonstandard models have infinitesimals, it will be quite challenging to give a concrete ...

**3**

votes

**1**answer

194 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 $\alpha$ ...

**12**

votes

**0**answers

503 views

### Arguments against Freiling's argument against Continuum Hypothesis

Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...

**1**

vote

**1**answer

274 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

**0**answers

114 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

**2**answers

370 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 '...

**9**

votes

**1**answer

423 views

### Examples of abstractions that did *not* turn out to be useful [closed]

I’ve read (but cannot find any reference now) that new abstract mathematical concepts like set theory and – not too long ago – category theory were in their time often considered too abstract to be ...

**53**

votes

**3**answers

4k views

### What was Hilbert's view of Gödel's Incompleteness Theorems?

According to Solomon Feferman, in his slide presentation "Three Problems for Mathematics", Hilbert wrote (in regards to Gödel's second incompleteness theorem):
...the end goal [is] to establish as ...

**2**

votes

**0**answers

108 views

### A question regarding an analogue of the Kleene $T$-predicate for Koepke's ordinal computability

Does Koepke's notion of ordinal computability admit an analogue of the Kleene $T$-predicate? If so, is the existence of such a $T$-predicate independent of $ZFC$? Also, if one assumes the existence ...

**2**

votes

**0**answers

73 views

### Which self-reference restrictions can be weakened in probabilstic logic?

This work suggests that there is some generalization of Truth in terms of probability, which can be definable within the logic itself.
Is where any other thorems on self-reference restrictions, which ...

**28**

votes

**7**answers

2k views

### Why should we believe in the axiom of regularity?

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/...

**0**

votes

**2**answers

321 views

### A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measurable cardnals

A major argument against Freiling's Axiom of Symmetry is the following (this from the wikipedia article of the same name):
"The naive probabalistic notion used by Freiling tacitly assumes that there ...

**3**

votes

**1**answer

270 views

### Information theory from negative probability

Szekely provides a convincing argument of negative probability here:
http://www.wilmott.com/pdfs/100609_gjs.pdf
What does a reformulation of classical information theory built from negative ...

**0**

votes

**1**answer

236 views

### A question regarding extendible cardinals and a result of M. Magidor

The following definitions and Theorems come from M. Magidor's paper "On the Role of Supercompact and Extendible Cardinals in Logic" (Israel J. Math., Vol. 10, 1971):
"Definition: Logic is called $\...

**4**

votes

**1**answer

278 views

### Plausibility argument for a measurable cardinal

The following question is not mathematically precise but perhaps of some philosophical interest.
A typical plausibility argument for assuming the existence of inaccessible cardinals goes as follows: ...

**1**

vote

**1**answer

347 views

### Is second-order ZFC categorical with regard to its proper class models

Second-order ZFC offers partial categoricity in the sense that, given any two models, one of them must be isomorphic to an initial segment of the other [1]. However, this leaves questions regarding ...

**32**

votes

**8**answers

5k views

### Uninteresting questions with interesting answers [closed]

What are best examples of questions in mathematics that are not interesting until one knows the answers, whose answers themselves are what is interesting?
The thing that prompts me to post this is ...

**17**

votes

**2**answers

1k views

### Philosophical arguments in defense (or against) large cardinals

The question is essentially what is asked in the title. I split it into two parts
(A) (Arguments supporting the existence of large cardinals) What are the main philosophical arguments in defense ...

**7**

votes

**1**answer

261 views

### Is $ACA_0$ + `True Arithmetic exists' interpretable in $ACA$?

Maybe someone here can help me with a question concerning second-order arithmetic. Consider the system $ACA_T := ACA_0 + \exists X \forall x (x \in X \leftrightarrow T(x))$, where $T(x)$ is a $\Pi_1^1$...

**14**

votes

**2**answers

979 views

### nonstandard models and mathematical theorems

Is there a first order statement about the natural numbers (not nonstandard analysis) such that the truth of the statement is easier to see in a nonstandard model? In other words, do nonstandard ...

**11**

votes

**2**answers

1k views

### Is there a compendium of the consistency strength between the most important formal theories?

Preliminar Notions:
A formal system is a tuple $(\Sigma,G,A,R)$ where $\Sigma$ is an alphabet (set of symbols), $G$ is a formal grammar on $\Sigma$ that generates a formal language $L$ (set of well ...

**2**

votes

**0**answers

248 views

### Interpretation of Shannon Entropy Application

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Let entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ be given
by $$H(\mathcal{A}...

**6**

votes

**3**answers

455 views

### Extensionality in HoTT versus extensionality in internal language of a category

What's the extension of judgmental identity in HoTT (homotopy type theory)?
The Martin-Löf intensional dependent type theory with identity types is called (definitionally) extensional if the ...

**79**

votes

**23**answers

13k views

### Has philosophy ever clarified mathematics?

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...

**8**

votes

**2**answers

1k views

### The impact of large cardinals in mathematics [closed]

What are the main applications of large cardinals in ordinary mathematics, and what is the philosophy behind using them. In particular:
Question 1. What is the philosophy behind accepting large ...

**12**

votes

**1**answer

840 views

### Time in Girard's Geometry of Interaction

Jean-Yves Girard writes at the end of his paper
"Towards a Geometry of Interaction", page 105, that we have three intuitions about the nature of time:
time is logic modulo the order of rules,
time ...

**-4**

votes

**1**answer

272 views

### Universal quantifier in Russell's Theory of descriptions - Who is the UNIVERSE? [closed]

To moderators: Please don't delete or migrate this thread: It's by no means PURE PHILOSOPHICAL, but mostly a mathematical logic question!
In Russell's 1905 paper "on denoting" in which he introduces ...