5
votes
1answer
78 views

Defining truth predicates in set theory

In this blog post J.D.Hamkins shows that KM set theory can define truth predicate for first-order set theory, which means, I believe, that there is a second-order definition of such predicate and KM ...
7
votes
1answer
82 views

Indiscernible subsequences

Work in a saturated model $\cal U$ of sufficiently large cardinality. Are there assumptions on $\kappa<|\cal U|$ that guarantee that any sequence $\langle a_i:i<\kappa\rangle$ has an ...
1
vote
1answer
105 views

Miller's Construction, Partition Principle and Failure of Axiom of Choice

Partition Principle ($PP$) is the following statement: For all sets $a$, $b$ there is an injection $f:a\rightarrow b$ iff there is a surjection $g:b\rightarrow a$ It is known that $ZF\vdash ...
1
vote
0answers
58 views

Are there axiomatizations of first order logic or set theory defined in first order logic or set theory?

There are several axiomatizations for number theory, group theory, and other theories represented in first order logic. Further, these theories are also representable in set theory such as $\sf ZFC$ ...
3
votes
1answer
49 views

What is the definition of the Feferman-Levy model?

Any (reference to) definition of Feferman-Levy model in set theory? I cannot find any... Though I know what is Levy collapse.
3
votes
0answers
67 views

Expository Papers on Extenders

Extenders are discussed in many set theory text books. Here I am looking for some expository "papers" which are focused on this subject and its connection with forcing and large cardinals. More ...
1
vote
0answers
95 views

A Question Regarding Consistent Fragments of Naive (Ideal) Set Theory

It is well known that Naive (to some, otherwise known as Ideal) set theory, that is, the set theory generated by the axioms: (EXT) (x)(x $\in$ A iff x $\in$ B) iff A=B (COMP) ($\exists$y)(x)(x ...
5
votes
1answer
140 views

cov(meager) strictly between $\aleph_1$ and $2^{\aleph_0}$

Is is consistent that $\aleph_1 < \text{cov(meager)} < 2^{\aleph_0}$? I can only seem to find references for results that assert it is consistent that it (or other cardinal characteristics) is ...
3
votes
2answers
86 views

Distinguishing sets according to more fine-grained notions than cardinality.

I'm interested in distinguishing sets according to more fine-grained notions than cardinality. Now I don't know a thing about computability theory, but it seems to me that considering sets up to ...
1
vote
1answer
83 views

Reference request, Descriptive set theory

I was wondering what a good text would be to learn descriptive set theory out of? Hopefully something more in the spirit of Kunen's text on the introduction to independence proofs.
2
votes
1answer
119 views

Best Less-Famous Texts for Forcing

There are many books, papers and lecture notes which give an introduction to forcing (e.g. Jech or Kunen's books) but here I am looking for some possibly less-famous useful comprehensive texts for ...
2
votes
1answer
57 views

Axiom of Limitation of Size Reference Request

On the wiki page for the axiom of limitation of size in NGB is states that the axiom of replacement and the axiom of global choice are equivalent to the axiom of limitation of size (see ...
4
votes
1answer
38 views

Extension of ZFC models preserves cardinals

Let $M \subseteq N$ be countable transitive ZFC set models. Assume that this extension preserves cardinals, i.e. if $\alpha$ is an ordinal number (this notion is absolute) such that $(\alpha \text{ is ...
5
votes
1answer
75 views

Partial order on cardinalities without the axiom of choice

Cardinality can still be defined without choice, e.g. as equivalence class of equipotent sets, see Defining cardinality in the absence of choice. Injections define partial order on cardinalities by ...
3
votes
2answers
93 views

Introduction to Proper Forcing Reference

What is a good introduction to proper forcing? I am aware of Shelah Proper and Improper Forcing, but I heard this book may be somewhat challenging to read. There is also Devlin's The Yorkshiremen's ...
1
vote
1answer
85 views

Modal set-theory

In his “The Potential Hierarchy of Sets”, Review of Symbolic Logic 6:2 (2013), 205-28 Øystein Linnebo has proposed a modal set-theory. I was wondering what kind of utility can such a theory have for a ...
3
votes
1answer
87 views

Has anyone considered axioms to the effect that: “The axiom of constructibility fails very very badly?”

If I'm not mistaken, the axiom of constructibility basically says that the universe has no (non-trivial) inner models. Has anyone considered axioms of the opposite flavour, basically asserting that ...
2
votes
0answers
125 views

Are there any good documentary films about the continuum hypothesis?

Are there any good documentary films about the continuum hypothesis? I'm looking for something slightly more serious than the usual "Cantor showed that infinity plus one equals infinity and then went ...
1
vote
1answer
78 views

Theorem of Galvin, Mycielski and Solovay

I don't know is this the right place to ask this question, but can someone tell me where I can find the proof of the theorem of Galvin, Mycielski and Solovay. Theorem that says that a subset $X$ of ...
5
votes
1answer
193 views

Asymmetric roles that are symmetric in every instance

This is similar to something else I posted, but this time we'll pretend we've never heard of infinite sets or infinite series. \begin{align} & \sin(\alpha+\beta+\gamma+\delta+\varepsilon+\zeta) ...
3
votes
1answer
132 views

Book recommendation in Foundational Mathematics

I have been navigating in this "foundational world" of mathematics for a while now ( but certainly not long enough and not deep enough ) and have read a bit about many different topics : set theory, ...
2
votes
1answer
69 views

Examples for intensional set theories.

The normal set theory of todays mathematics (ZFC) is extensional, i.e. it has the axiom of extensionality $$\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \Rightarrow A = B)$$ Are ...
3
votes
1answer
68 views

Book/Books leading up to the the axiom of choice?

I am familiar with the axioms of ZF set theory and some basic uses of them to completely formally construct more complex objects such as natural numbers etc. However I have pretty much no background ...
3
votes
1answer
114 views

Recommended books/articles for learning set theory

What is the recommended reading for thoroughly learning set theory? I'm currently studying Kunen's book [1]. But what then, and in what order? One needs to learn large cardinals, inner models and ...
1
vote
0answers
57 views

Subject-level guide for Princeton Companion to Math?

I have the Princeton Companion to Mathematics, which I'm enjoying overall. However, right now it's a lot more useful to me for expanding on topics I'm already somewhat familiar and less useful for ...
4
votes
2answers
124 views

The regularity of successor cardinal

I was looking at two different proofs of the fact that successor cardinals are regular. It struck me as odd that both proofs used AC. Looking at the concepts involved in defining cofinality I feel as ...
1
vote
1answer
77 views

Reference Request Scott's Trick

Does anyone know of a reference for Scott's Trick. I can't find it in Set Theory-Jech?
12
votes
2answers
201 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 ...
8
votes
0answers
132 views

Cantor-Schröder-Bernstein without elements

The Cantor-Schröder-Bernstein Theorem for the category of sets has several "analogues" for other categories as well, for example measurable spaces and operator algebras (but not for arbitrary ...
11
votes
1answer
141 views

Consistency strength of Turing measurability

This is probably well-known to recursion theorists, but as google didn't help me, I'll ask it here. Convention: All sets of reals in the following discussion are assumed to be closed under Turing ...
2
votes
2answers
53 views

Reference Request for GB-set theory

Could anyone give me a reference for a book which has an introduction to set theory from the GB axioms as opposed to ZFC, everything I read seems to just look at things from ZFC (Jech...) Thanks for ...
4
votes
2answers
160 views

Set theory aspects of category theory

I have never learnt axiomatic set theory, but have studied it from Munkres's Topology book first chapter. So I do not understand the difference between a class and a set except in some vague sense. ...
4
votes
1answer
85 views

Has anyone successfully axiomatized the category of finite sets? In such a way that the resulting theory is bi-interpretable with PA.

In studies of ZFC, it is conventional to take Peano arithmetic (hereafter PA) as the metatheory. However, I don't like this convention; I think a better approach would be a metatheory (like ZF-fin ...
2
votes
0answers
45 views

A general form of Tukey's Lemma

In the book "General Topology" by John Kelley, Tukey's Lemma is stated as: "If a family of sets is of finite character, it has a maximal member". In the end of the section on set theory, there is the ...
4
votes
1answer
127 views

Introducing ordered pairs in an axiomatic way [duplicate]

Suppose that in $ZFC$ we have introduced ordered pairs not in the usual way as $(a,b) = \{\{a\}, \{a,b\}\}$ but axiomatically, by extending $ZFC$ by adding to $ZFC$ a new binary functional symbol $g$ ...
5
votes
1answer
121 views

Are there mathematical (combinatorial) objects to represent such set systems?

Background: This is a problem arising from my study on computer science. In its appearance, it involves set systems. A set system $\mathcal{S} = \{S_1, S_2, \ldots, S_m \}$ is a collection of ...
2
votes
0answers
67 views

Forcing Language

Does anyone know of a reference that explains the concept of forcing by fixing a forcing language that has a (I believe unary) predicate and does not mention the forcing poset? For example, Kunen's ...
4
votes
1answer
87 views

Are there such things as non-extensional set theories?

I have always assumed that extensionality is a paradigmatic example of a property of mathematical objects (sets) which is essential to those objects--- if your set theory doesn't obey extensionality, ...
0
votes
1answer
67 views

A few questions about the notion of 'proper class' in set theory

In a previous discussion on this website, Asaf Karagila said something to the effect of: Given a model $(V,\in')$ of ZFC, the classes of that model are (by definition) the subsets of $V$ definable ...
3
votes
1answer
62 views

Con ZF implies Con ZFC using set sized models

Can we use forcing to construct models of ZFC and ZFC + GCH starting from c.t.m s of ZF? The usual way to obtain the associated relative consistency results (Con ZF implies Con ZFC and Con ZF implies ...
5
votes
3answers
195 views

Foundations of Forcing

I am currently studying Forcing methods in order to understand some independence results and model's constructions. Now I am interested on formalizing the main notions around forcing such as ...
8
votes
1answer
191 views

Can we found mathematics without evaluation or membership?

In some sense, composition generalizes evaluation. The trick is, instead of writing $f(x)$ for $x$ an element of the domain of $X,$ we write $f \circ x$ for $x$ a function $1 \rightarrow X$. ...
7
votes
0answers
262 views

Proving equivalence of a tree-based version of Countable Choice for families of finite sets.

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of ...
3
votes
1answer
110 views

Dependent choice does not imply “the reals are well-ordered”; citation?

As silly as this sounds, I can't find a proof that the axiom of dependent choice (DC) does not imply that the reals are well-orderable. My memory is that this is a fairly early result in the history ...
0
votes
1answer
88 views

Dependence of Dedekind's Theorem on AC using Scott's trick

I would like some reference about this fact: the essential dependence on choice of Dedekind's theorem in, e.g., Kelley-Morse set theory using Scott's trick.
4
votes
5answers
528 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 ...
3
votes
2answers
171 views

Random reals and Martin-Löf randomness

My questions are about the relationship between the following notions of randomness: A real $r$ is random over the model $M$ if $r\notin B$ for every null Borel set $B$ coded in $M$. A real $r$ is ...
2
votes
1answer
193 views

Modern reference on logic-set theory-foundation

I'm looking for a modern book on logic-set theory-foundation written as the Bourbaki's set theory. I'm particularly interested in a formal exposition of ZFC axiom with logic-set Grothendieck universe. ...
2
votes
1answer
82 views

Are the closed ordinals (apart from $0$ and $1$) precisely the regular cardinals?

Given a partially ordered set $P$, a collection of partially ordered sets, call it $\mathcal{Q}$, and an arbitrary function $f : P \rightarrow \mathcal{Q},$ we can form a new partially ordered set ...
3
votes
2answers
144 views

Good resource to learn transfinite induction and/or recursion?

I'm currently reading John H. Conway's On Numbers and Games, but without a good understanding of transfinite induction and/or recursion, progress is very slow. What's a good resource for learning ...