-2
votes
0answers
87 views

Absoluteness of $\forall x (x=x)$

Is there any kind of (set-theoretic) absoluteness result for the formula $\forall x (x=x)$? And what about for $\exists x (x=x)$? I know $x=x$ is absolute given that it's $\Delta_0$. Also, if I'm ...
0
votes
2answers
116 views

Is the “Most Important Property a Set S has” Necessary and Sufficient to Define a Paradox-Free Notion of Set?

About a year and a half ago, while I was looking on the Web for papers regarding the Russell paradox, I chanced to find an interesting concept. This concept was contained in what (for want of a ...
3
votes
2answers
153 views

What is the meaning/purpose of finding the “foundations of mathematics”?

I've read in a lot of places how there was a "foundational crisis" in defining the "foundations of mathematics" in the 20th century. Now, I understand that mathematics was very different then, I ...
2
votes
2answers
153 views

Is it Theoretically Impossible to Demonstrate that Set Theories Are Consistent?

I have to present on the main realist and non-realist arguments for/against set theory. According to one of my sources, it remains a matter of debate as to whether any of the set theories' (ZF, NF, ...
6
votes
2answers
249 views

Set theoretic realism

What are the main contemporary arguments for and against realism about set theory?
1
vote
1answer
93 views

A question regarding Worldly Cardinals and L

For some $L_\kappa$ in the constructible hierarchy, does there exist a $\kappa$ such that $\kappa$ is a worldly cardinal and that $L_\kappa$ contains all of the constructible reals? The motivation ...
2
votes
1answer
54 views

Difference between impredicative and predicative version of separation axiom

What is the difference between an impredicative and a predicative version of the separation axiom in ZFC: $$\forall x \exists y \forall z ( z\in y \leftrightarrow (z \in x \wedge \phi (z)) $$ What ...
4
votes
3answers
146 views

Are there other approaches for the foundations of mathematics, other than logic and set theory?

Are there other approaches for the foundations of mathematics, other than logic and set theory? And why does set theory begin talking about objects and groups of objects. Is it proven somewhere that ...
1
vote
2answers
89 views

Consistency of ZFC and the key assumption [closed]

I recently read this answer to a MathOverflow question that got me thinking. Very roughly, here's what the author of that answer says: Gödel's second incompleteness theorem implies that if there ...
6
votes
1answer
202 views

New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: $Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in ...
0
votes
0answers
63 views

A Question Regarding Ordinal Turing Machines

Consider the following theorem of Koepke: 'A set x of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of the constructible universe L". Taking ...
5
votes
2answers
79 views

What is gained by internalizing LST (the language of set theory)?

I'm reading up on Gödels constructible universe L in the book "Constructibility" by Devlin, and by comparing his text with texts like Kunen and Jech, there is one thing in particular that he's doing ...
12
votes
2answers
199 views

Founding Arithmetic on geometry

In the past I found some fleeting references that some (Frege in his later years being one of them) tried to found arithmetic not on set-theory and logic but on geometry and logic. Unfortunedly Frege ...
3
votes
2answers
125 views

Non-well-founded models viewing well-founded models as non-well-founded.

I'm currently thinking about how different models of set theory view each other. In particular I'm looking at how well-foundedness behaves between different models. So we have the Axiom of ...
6
votes
1answer
125 views

Do second-order categoricity proofs require a background concept of set?

In his article "The Set-Theoretic Multiverse", Joel David Hamkins (as part of his reply to Donald Martin's argument that the set-theoretic universe is unique, found in "Multiple Universes of Sets and ...
1
vote
0answers
63 views

Are there intensional classes independent of the set universe?

The hereditarily finite sets can be regarded as purely extensional sets. Furthermore, they are quite independent of the underlying set universe (at least if we look at them from an extensional point ...
3
votes
2answers
136 views

Understanding: Axiom of Specification and Russell's Paradox: there is no universe?

Following Halmos's Naive Set Theory, Russell's Paradox emerges from using the axiom of specification (that for every set $A$ and property $\phi$ there exists a set $Y$ whose elements are those ...
19
votes
1answer
442 views

Is there any mathematical meaning in this set-theoretical joke?

Recently I heard a joke: If an object exists, mathematicians call it a set and study it. But if an object does not exist, mathematicians call it a proper class and study it anyway. I wonder, ...
4
votes
5answers
491 views

Books on logic, proof theory and set theory?

I graduated in Computer Science at University of Bologna in Italy some years ago. For various reasons now I am discovering a back interest in mathematic logic higher than I was a student. I have only ...
4
votes
1answer
157 views

Does ZFC have an intended interpretation?

I know that PA has an intended interpretation, namely $\mathbb{N}$, and the usual axioms of the real line have an intended interpretation, namely $\mathbb{R}$. Does ZFC have an intended ...
6
votes
2answers
277 views

Is the proper class of all ordinals equivalent to the potential infinity of pre-Cantor times?

My understanding is that the class of all ordinals is, by definition a proper class. This in the end is done to avoid a paradox: the collection of all sets would be paradoxical if you allow it to be a ...
2
votes
3answers
198 views

Subsets as non-mathematical objects?

I think of mathematical objects as individual things that exist by their own (either abstractly or concretely) and can be represented mathematically. When thinking of subsets, I'm in doubt if ...
2
votes
1answer
188 views

Did large cardinals exist before 1963?

I'm curious to know the history of the interaction between large cardinals and traveling to (creating) universes through forcing. The question arose because I understand that Peano Arithmatists ...
7
votes
1answer
152 views

Set theoretic implications of constructions in Differential Geometry/ Topology

In subjects like Differential Geometry/ General Topology one often constructs for each $x$ in a space $X$ a set $U_x$ satisfying certain properties. Examples where one does constructions like this: ...
15
votes
1answer
351 views

Are there areas of mathematics (current or future) that cannot be formalized in set theory?

I often read that ZFC can formalize "most" of everyday mathematics, but I could never find an example which it cannot. The closest I got is differential geometry (DF), where some article mentions that ...
0
votes
0answers
106 views

Does it become more likely that ZFC is consistent, the more time we explore it without finding a contradiction?

Intuitively, the more time we spend exploring ZFC without finding a contradiction, the higher the (subjective) probability that ZFC is consistent. Is this intuition sound? If not, why not?
2
votes
2answers
183 views

What does Russell mean when he defines the “Posterity… with respect to the immediate predecessor”?

The the Introduction to Mathematical Philosophy, Russell defines the "posterity" of a given number with respect to the relation "immediate predecessor" as all those terms that belong to every ...
8
votes
4answers
604 views

What would happen if ZFC were found to be inconsistent?

If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that ...
1
vote
1answer
79 views

A Simpler Characterization of Inductive Definitions?

While reading appendix A of John Harrison's "Handbook of Practical Logic and Automated Reasoning" a somewhat advanced theorem is appealed to as a prerequisite for characterizing when an inductive ...
8
votes
6answers
1k views

Why accept the axiom of infinity?

According to my readings, Russell showed that a principle Frege used to reduce Peano arithmetic to logic lead to a contradiction. So, Russell tried to reduce mathematics to logic a different way but ...
2
votes
1answer
287 views

Russell Paradox and set theories

The Russell paradox arise in the Cantor set theory, but it can be avoided in the $ZF$ and in $NGB$ axiomatic set theory. Are there other axiomatic set theories in which this paradox can be avoided? ...
3
votes
0answers
183 views

Does the concept of predicativity need to be formalized to go beyond Feferman-Schutte ordinal?

Feferman-Schütte ordinal is sometimes said to be: ....first impredicative ordinal, though this is controversial, partly because there is no generally accepted precise definition of "predicative". ...
10
votes
3answers
592 views

Difference between undecidable statements in set-theory and number theory?

Do all statements about the integers have a definite truth value? For instance: Goodstein's theorem is clearly true, otherwise we could find a finite counterexample thus it would be possible to ...
22
votes
6answers
2k views

What are natural numbers?

What are the natural numbers? Is it a valid question at all? My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational ...
8
votes
4answers
446 views

Consequences of solving the Halting problem

What impact would a device (ie super-computer or relativistic computer or other method) that solves the halting problem have on math? Would there be any mathematical problems left to solve? What ...
0
votes
2answers
352 views

Equality of abstract structures

Philosophical questions concerning the difference between equality, isomorphism, equality upto (unique) isomorphism, undistinguishability, and the like are not very popular among practicing ...
28
votes
6answers
2k views

If all sets were finite, how could the real numbers be defined?

An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who ...