All Questions
Tagged with set-theory reference-request
471 questions
4
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3
answers
249
views
An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
$\newcommand\R{\Bbb R}$Assuming the axiom of choice, is there an uncountable Lebesgue-measurable subset $S$ of $\R$ that contains no nonempty perfect set?
Of course, such a set $S$, if it exists, ...
- 128k
11
votes
1
answer
369
views
Reference request: The non-productivity of Lindenbaum numbers
For a set $X$, the Lindenbaum number of $X$, $\aleph^*(X)$, is the least non-zero ordinal $\alpha$ such that there is no surjection $X\to\alpha$. It seems to be well-known that for infinite sets $X$ ...
- 2,206
5
votes
0
answers
158
views
If $\omega_1$ is not inaccessible in $L$, how hard can it be to find a non-measurable $\Sigma^1_3$ set of reals?
In his wonderfully titled paper Can you take Solovay's inaccessible away? Shelah showed that if every $\mathbf{\Sigma}^1_3$ set of reals is Lebesgue measurable, then $\omega_1$ is an inaccessible ...
- 12.5k
6
votes
0
answers
188
views
Is there a characterization of measurables in terms of indiscernibles?
There is a characterization of $\alpha$-Erdős cardinals in terms of sets of indiscernibles of order type $\alpha$. There is also a characterization of Ramsey cardinals in terms of sets of good ...
- 2,031
7
votes
3
answers
769
views
Implicit uses of Countable or Dependent Choice
What are instances of implicit reliance on countable or dependent choice in classic books? Two examples are
Introduction to Commutative Algebra by M.F. Atiyah and I.G. MacDonald
where it is claimed,...
11
votes
1
answer
500
views
Uncountable families of measurable sets with pairwise positive intersections
Let $(X,\mathcal{B},\mu)$ be an arbitrary finitely additive probability measure space, let $a>0$ and let $(A_i)_{i\in I}$ be an uncountable family of subsets with measure $\geq a$.
Is there an ...
- 10.6k
7
votes
0
answers
142
views
What is the forcing $\bf U$ from Bartoszyński-Judah?
In Set Theory - on the structure of the Real Line by Bartoszyński & Judah, a forcing notion $\bf U$ is mentioned on page 339, allegedly corresponding to $\rm{cof}(\cal N)$ as it has several ...
- 393
3
votes
0
answers
287
views
What did Mirimanoff say about Intuitionism?
Dmitry Mirimanoff, "L'intuitionisme", Alma Mater n° 6, Geneva, 1945.
Most of Mirimanoff's work was in number theory, but he wrote three papers about set theory that were way ahead of their ...
- 8,481
16
votes
2
answers
1k
views
CH in non-set theoretic foundations
I asked this question on stack exchange and got little attention, barring a nice example I intend to look into. The original post can be found here: https://math.stackexchange.com/q/4941233/1053681
I ...
- 261
9
votes
0
answers
274
views
Has a computer search for inconsistency of large cardinals been carried out before?
In discussion about the consistency of large cardinals, a justification which occasionally appears is that despite years of work, no inconsistency has currently been discovered. For example, here are ...
- 2,031
6
votes
1
answer
349
views
Where is the original theorem shooting a club to kill a Mahlo cardinal?
I just want to make sure that I have the correct reference for the original theorem of shooting a club of singular cardinals to make a Mahlo cardinal become a non-Mahlo inaccessible cardinal. I can't ...
- 163
4
votes
0
answers
143
views
Part II to Ketonen's "Set Theory for a Small Universe I. The Paris-Harrington Axiom"
There is an unpublished manuscript "Set Theory for a Small Universe I. The Paris-Harrington Axiom" by Ketonen which appeared early in the study of the Paris-Harrington theorem, around 1979. ...
- 2,031
11
votes
1
answer
1k
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Had this attempt to salvage naïve comprehension been studied before?
Is the following a possible way to overcome inconsistency with naive comprehension:
We add an $\in_n$ symbol for each natural $n$ to the signature of this theory, which is a first order theory with ...
- 11.3k
4
votes
0
answers
149
views
Computable subsets of non-standard models of arithmetic
By Tennenbaum's theorem, there exists no computable non-standard model of $\mathsf{PA}$. That is, for any nonstandard model $M$, we cannot define an encoding of the integers $x\in M$ such that $+_M$, $...
- 291
7
votes
0
answers
349
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An open set which is not the union of a closed set and a countable set
The following fact is probably a known result:
Fact. Let $X$ be an uncountable Polish space. Then there exists an open subset of $X$ which is not the union of a closed set and a countable set.
Proof:...
- 1,511
6
votes
1
answer
199
views
$L(\mathbb{R})$-absoluteness from a proper class of Woodins: source?
For a paper I'm writing I need to use (as a blackbox) the following theorem: if there is a proper class of Woodin cardinals and $G$ is set-generic, then $L(\mathbb{R})$ and $L(\mathbb{R})^{V[G]}$ are ...
- 21.2k
9
votes
0
answers
168
views
Algebraic structures on spaces of ultrafilters
The space of ultrafilters on $\omega$ has a natural semigroup structure, and ultrafilters that are idempotent in that algebra have seen applications in combinatorics on the natural numbers, for ...
- 18.6k
15
votes
2
answers
1k
views
Proof/Reference to a claim about AC and definable real numbers
I’ve read somewhere on this site (I believe from a JDH comment) that an argument in favor of AC (I believe from Asaf Karagila) is that without AC, there exists a real number which is not definable ...
- 293
9
votes
2
answers
455
views
Determinacy and Woodin cardinals
I am looking for a reference for the following result:
Theorem Assume $\kappa$ is a Woodin cardinal. Then after forcing with the Levy collapse $\mathbb{P}=Col(\omega, \kappa)$, the $\Sigma^1_2$-...
- 32.1k
10
votes
1
answer
515
views
Earliest proof of Solovay's theorem for successor cardinals
Solovay's partition theorem states that a stationary set over a regular cardinal $\kappa$ can be partitioned into $\kappa$-many disjoint stationary sets. The full theorem was proven by Solovay in 1971 ...
- 576
6
votes
0
answers
144
views
Alternative proofs of the countable chain condition in forcing
Advance warning: This question is more about history and pedagogy than "hard" mathematics.
I am studying Cohen forcing with the forcing poset $(\operatorname{Fin}(E,2),\supseteq,0)$, and I ...
13
votes
0
answers
2k
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Is there any correspondence between Gödel and Kreisel that supports Kreisel's observation that Gödel changed his mind about his 1938 set theory note?
At a conference in 1965 there were some interesting comments made by Kreisel and Mostowski asserting that Gödel later changed his mind regarding his1938 note on his set theory results (see Problems in ...
- 481
6
votes
1
answer
227
views
Forcing $\neg\square_{\omega_1}$ from a Mahlo cardinal, reference
In Jensen's The fine structure of the constructible hierarchy, it is stated that Solovay proved the consistency of $\neg\square_{\omega_1}$ by collapsing a Mahlo cardinal to $\omega_2$. I was ...
- 2,286
5
votes
0
answers
150
views
Consistency upper bounds for $\neg\square_{\aleph_\omega}$
In the introduction of Cummings and Friedman's $\square$ on the singular cardinals the following is written:
Failure of $\square_\lambda$ for $\lambda$ singular is stronger and rather more ...
- 2,286
4
votes
1
answer
204
views
How probability-rich is the $\sigma$-algebra generated by a sequence of sets? (Sierpiński's theorem on non-atomic measures without using the AoC.)
$\newcommand\F{\mathcal F}\newcommand\si{\sigma}\newcommand\Om{\Omega}\newcommand\ep{\varepsilon}$Let $p\in(0,1)$ and let $(\Om,\F,P)$ be a probability space. Let $(A_n)$ be a sequence in $\F$ such ...
- 128k
4
votes
0
answers
165
views
Looking for reference on Vopenka's theorem on generic extensions of HOD
Chapter 15 of the third edition of Jech's textbook on set theory gives Vopenka's theorem as saying that if $V=L[A]$ where $A$ is a set of ordinals then $V$ is a set generic extension of $HOD$, whereas ...
- 2,125
4
votes
1
answer
148
views
Reference request: $\kappa^+$-saturated ideal from iterated Cohen forcing on measurable $\kappa$
In Kunen [1] the author makes the following note: Let $\kappa$ be measurable with normal measure $\mathscr{U}$ in a model of $\mathsf{GCH}$. Let $\mathbb{P}$ be an iteration of $\operatorname{Add}(\...
- 2,206
8
votes
0
answers
368
views
An obscure case of Curry-Howard
It is a theorem of the Intuitionistic Propositional Calculus that
$$
(p\to q)\to p = (q\to p) \land ((p\to q)\to q).
$$
The Curry-Howard correspondence realizes this as a pair of operators (for any ...
- 18.4k
4
votes
2
answers
275
views
Diagrammatic representation of sets as irregular plane figures
I am trying to find the earliest use of plane diagrams of various shapes for representing sets. For example, this is snapshot is from the book called Some Modern Mathematics for Physicists and Other ...
- 879
6
votes
1
answer
174
views
Strengthening of a classical set mapping theorem of Lázár
We are writing a paper in which we need to use the following theorem to prove that a certain poset satisfies c.c.c. As far as I remember I learned this theorem from András Hajnal.
Theorem 1: If $\...
- 1,467
6
votes
1
answer
412
views
Second-order ordinal definability
As is familiar, a set $S$ is ordinal definable ($S \in \mathsf{OD}$) just in case there exists a formula $\varphi(x, \vec{z})$ of the first-order language of set theory with parameters $\vec{z} = z_1, ...
- 1,155
10
votes
1
answer
416
views
Consistency strength of strongly compact cardinal
Where can I find a proof that strongly compact cardinal has higher consistency strength than Woodin cardinal, or even just strong? Recall that a strongly compact cardinal itself may not be strong, ...
- 333
5
votes
0
answers
213
views
Friedman's proof of covering lemma for $L$
There is a two-page proof of the covering lemma for $L$ using $\Sigma_n$ substructures (Theorem 3.10) in Sy Friedman's Fine Structure and Class Forcing, compared to the proof that spans about twenty ...
- 333
2
votes
2
answers
233
views
Name for a certain type of cardinal
I'm not a set-theorist, but I hope this question is appropriate. This is just a question about names:
Fix a cardinal $\lambda$. I'd like to know if there is a name for regular cardinals $\kappa$ such ...
- 15.8k
8
votes
1
answer
241
views
A reference for forcing projections
The idea of a projection $\pi\colon\mathbb{Q}\to\mathbb{P}$ of forcing notions is something like a combinatorial stand-in for the fact that forcing with $\mathbb{Q}$ produces a generic for $\mathbb{P}$...
- 2,389
12
votes
1
answer
376
views
Partition into antichains
I've read that the following statement is a result of Balcar, but I am unable to find a reference or a proof:
Theorem: If $\kappa\ge \lambda$ are infinite cardinals, then $[\kappa]^{<\lambda}$ can ...
- 1,467
4
votes
1
answer
208
views
Generic absoluteness
In Theorem 14 of "On The Question Of Absolute Undecidability" Peter Koellner describes a generic absoluteness result which could be summed up as $\Sigma^2_1(\Gamma^\infty)$-generic ...
- 2,125
5
votes
2
answers
432
views
Models of second-order arithmetic closed under relative constructibility
I know little to nothing about second-order arithmetic and its subsystems. However, I would like to understand when a model of (a subsystem of) second-order arithmetic ($\mathsf{Z}_2$) is downward ...
- 2,286
5
votes
0
answers
191
views
Reference-Request: Had this replacement principle been investigated before?
Replacement$^*$: If $\varphi$ is a formula with at least two variables occuring free, in which $``x,y"$ do not occur free, then:
$$\forall a \forall b \forall c \forall d \ ( \varphi(a,b) \land \...
- 11.3k
6
votes
1
answer
156
views
Preservation of cardinals implies preservation of cofinalities when $V=L$?
Kunen mentions the result stated in the title. It would be much appreciated if someone can give a reference for this. Thank you!
- 105
8
votes
0
answers
196
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Reference request: choiceless cardinality quantifiers
There is a substantial literature on the logic of cardinality quantifiers. (E.g., the quantifier $Q_\alpha$ where $M \vDash Q_\alpha x \, \varphi (x)$ iff $\vert \{a \in M : M \vDash \varphi[a] \} \...
- 1,155
7
votes
2
answers
440
views
On the existence of a real which is not set-generic over $L$
Recall that a real $r$ is set-generic over $L$ if there is a constructible forcing notion $\mathbb{P}$ and some $L$-generic filter $G\subset\mathbb{P}$ such that $r \in L[G]$.
I know that Jensen's ...
- 2,286
5
votes
0
answers
136
views
When does an iteration not add functions $\eta\to V$ at the final stage?
I am interested in better understanding the following property:
Let us say that an iteration of forcings $\langle\mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid\alpha\leq\gamma,\beta<\gamma\rangle$ is ...
- 2,206
2
votes
1
answer
600
views
"Potency set" for power set?
Cross-posted at HSM.
Has the term "potency set" been used in English language mathematics for power set, and, if so, what are good references?
It is relevant that for historical reasons, &...
- 3,574
15
votes
3
answers
2k
views
May two Cohen reals collapse cardinals?
My question is the following:
Let $M$ be a c.t.m. of $\mathsf{ZFC}$. Are there two reals $r_0,r_1 \in \mathbb{R}$ such that $r_i$ is Cohen over $M$ for $i=0,1$ and such that $\omega_1^M$ is countable ...
- 2,286
6
votes
1
answer
431
views
A strange product forcing
Given two forcing notions $\mathbb{P}_0,\mathbb{P}_1$, we know that the product $\mathbb{P_0}\times\mathbb{P_1}$ will produce the following diagram of models ordered by inclusion:
where $M$ is the ...
- 2,286
43
votes
4
answers
5k
views
Lists as a foundation of mathematics
I am wondering if there is a foundation of mathematics where not sets or "set-like objects" (such as objects of a suitable topos as in ETCS) are the primitive notion, but rather lists. These ...
- 63.1k
7
votes
1
answer
579
views
Progress on determining which partial orders embed into the rationals
The following result is relatively well-known: (for example in Math StackExchange answer #37161)
For every countable linear order $(R,\prec)$, there is an $X\subseteq\mathbb Q$ such that $(R,\prec)$ ...
- 2,031
5
votes
1
answer
135
views
References for the axiom of surjective comparability
The axiom $W_\kappa$, for $\kappa$ a cardinal, is the statement that for all sets $X$, either $|X|\leq\kappa$ (that is, there is an injection $X\to\kappa$) or $\kappa\leq|X|$. Is there literature on ...
- 2,206
7
votes
1
answer
357
views
Forcing axiom for a single poset
Let $FA_\kappa (\mathbb{P})$ be the claim that for every family $\mathscr{D}$ of dense sets in the poset $\mathbb{P}$ with $\vert \mathscr{D} \vert = \kappa $ there is a filter $G$ such that for all $...
- 505