# Tagged Questions

**5**

votes

**2**answers

788 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 ...

**5**

votes

**3**answers

702 views

### Why isn't there more interest in “large powerset axioms”?

By a large powerset axiom, let us mean informally an axiom that says that for some cardinal numbers $\kappa$, we have that $2^\kappa$ is somehow "large" or "difficult to access from below." For ...

**2**

votes

**2**answers

455 views

### Have axioms / axiom schemata of this flavour been proposed or otherwise considered?

With the exception of a few miscellaneous cases, the axioms (and/or schemeta) of ZFC can roughly be divided into two kinds:
Those that guarantee the existence of more complicated sets, given that ...

**22**

votes

**4**answers

963 views

### What is the definition of a large cardinal axiom?

In different books one can find different implicit definitions for a large cardinal axiom.
My question is that which one of these definitions are more popular or standard amongst set theorists?
Any ...

**18**

votes

**2**answers

2k views

### Where are Georg Cantor's Original Manuscripts?

Georg Cantor is famous for introducing transfinite numbers and set theory.
A main part of his mathematical point of view about this new type of "numbers" and this new "realm of mathematics" cannot be ...

**3**

votes

**1**answer

281 views

### Is “ultracompact” taken?

Almost-huge cardinals are characterizable in terms of coherent towers of supercompactness measures, with a certain property of the direct limit model (see Kanamori's book). A useful large cardinal ...

**4**

votes

**1**answer

875 views

### Preparing for Set Theory Research

Is reading Jech's text on Set Theory too little, just enough, or overkill to prepare oneself to do independent research in set theory? This would be my first attempt at doing independent research ...

**9**

votes

**3**answers

806 views

### Is there an observer dependent mathematics? [closed]

Is there any field of mathematics that deals with the role of the observer? E.g., some formulation in which a set is changed, in some unspecified way, when it is observed? Or maybe some philosophy of ...

**0**

votes

**2**answers

380 views

### Correct definition of the sequence of natural numbers with set theory, but without counting or measuring size [closed]

This question may appear banal, but there seems to be more than meets the eye; a common glitch is to explain numbers by the "size" of sets without saying how to measure or compare the size of sets.
...

**2**

votes

**0**answers

247 views

### Descriptive set theory on $\mathbb{R}^\mathbb{N}$

The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...

**3**

votes

**0**answers

237 views

### A Question Regarding Boolean-valued Models

What were the intuitions motivating the creation (or discovery, if you will) of Boolean-valued models? I have searched for the Scott-Solovay paper on the subject, but to no avail. There also seems to ...

**0**

votes

**2**answers

534 views

### What constitutes the division between discrete and non-discrete math? Are there any math subject where it's being blurred? [closed]

I often heard about this division but always in a non-formal manner. What constitutes it? Is it a limit operation? Or a fundamental distinction between countable and uncountable sets? And what math ...

**6**

votes

**2**answers

702 views

### Road to Solovay's Land.

In the first semester of 2012 I took a course in General Topology and Set Theory, at undergraduate level. For topology, I was instructed to use Engelking's General Topology; albeit I had a great ...

**8**

votes

**1**answer

754 views

### Hausdorff and Naive Set Theory

Erhard Scholz, in his article "Felix Hausdorff and the Hausdorff edition" writes the following:
"Hausdorff considered the contemporary attempts to secure axiomatic foundations for set theory as ...

**1**

vote

**1**answer

891 views

### Shortest formal statement equivalent to the continuum hypothesis

What is the shortest formal statement you can write that is provably equivalent to the Continuum Hypothesis in ZFC?
Please use only variables and the following symbols: $\forall, ...

**1**

vote

**1**answer

360 views

### Is there a countable pseudocharacter Hausdorff spaceļ¼such that…?

Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two ...

**6**

votes

**5**answers

738 views

### the example of ccc but not separable

I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance.
...

**8**

votes

**1**answer

463 views

### What is the etymology of zero-sharp?

I have wondered for a while what gave rise to the notation $0^\sharp$. According to wikipedia this is due to Solovay in 1967, but (perhaps unsurprisingly) there's no discussion of why that notation ...

**17**

votes

**4**answers

1k views

### Are proper classes objects?

Many of us presume that mathematics studies objects. In agreement with this, set theorists often say that they study the well founded hereditarily extensional objects generated ex nihilo by the ...

**9**

votes

**2**answers

590 views

### Why is this theorem (about $L(P(\omega_1))^V$ and $L(P(\omega_1))^{V[G]}$) nice?

I was recently told that the following (due to M. Viale) is a nice theorem:
Suppose there are arbitrarily large supercompacts, and $\mathrm{MM}$ holds in $V$. Let $G$ be generic for a proper ...

**7**

votes

**1**answer

500 views

### Explicit uses of alephs above 'small ones'

In a paper placed on the arXiv today Shelah references theorem 0.9 from this paper (also Shelah) that uses $\aleph_{736}$ as an upper bound. This strikes me as analogous to Skewes' number. Are there ...

**14**

votes

**2**answers

2k views

### Should there be a true model of set theory?

As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...

**31**

votes

**7**answers

5k views

### Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...

**35**

votes

**5**answers

3k views

### What's wrong with the surreals?

Of all the constructions of the reals, the construction of the surreals seems the most elegant to me.
It seems to immediately capture the total ordering and precision of Dedekind cuts at a ...

**18**

votes

**9**answers

4k views

### Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]

As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is ...

**29**

votes

**5**answers

3k views

### Why do categorical foundationalists want to escape set theory?

This is a question that I have seen asked passively in comments relating to the separation of category theory from set theory, but I haven't seen it addressed in full.
I know that it's possible to ...

**27**

votes

**4**answers

3k views

### Is “all categorical reasoning formally contradictory”?

In the December 2009 issue of the newsletter of the European Mathematical Society there is a very interesting interview with Pierre Cartier. In page 33, to the question
What was the ontological ...

**8**

votes

**4**answers

1k views

### the delta system lemma outside set theory

The lemma:
Any uncountable set $S$ of finite sets has an uncountable subset $\Delta \subseteq S$ and an $x$ such that $\forall a,b \in \Delta$, if $a \neq b$ then $a \cap b = x$. $\Delta$ is called a ...

**9**

votes

**8**answers

2k views

### Is there a ground between Set Theory and Group Theory/Algebra?

It is well known that there are strong links between Set Theory and Topology/Real Analysis.
For instance, the study of Suslin's Problem turns out to be a set theoretic problem, even though it started ...

**-2**

votes

**1**answer

307 views

### Properties of collections (functions) that make them proper classes (uncomputable)

There are collections too big to be a set, e.g. the collection of all sets (in ZFC), and there are collections that cannot be sets for "pure" logical reasons, e.g. the collection of sets that do not ...

**6**

votes

**2**answers

501 views

### measure theory for regular cardinals

Measure theory is somewhat focused on the cardinal $\aleph_0$: First of all we have the usual $\sigma$-additivity, Polish (separable!) spaces such as $\mathbb{R}^n$, countable sequences and limits, ...

**4**

votes

**6**answers

901 views

### Are there nonequivalent randomnesses?

There are nonequivalent geometries, nonequivalent groups finite and infinite, nonequivalent logics ( fregean and nofregean http://www.formalontology.it/suszkor.htm), even nonequivalent logicians;-)
...

**1**

vote

**3**answers

2k views

### “Interesting” properties of sets of natural numbers

On Wikipedia, there is a list of properties of sets of reals, which are in some sense "interesting": just have a look.
I could not find a comparable list of properties of sets of natural numbers ...

**5**

votes

**5**answers

3k views

### Models of ZFC Set Theory - Getting Started

For just any first-order theory: What are the sets I am supposed/allowed to think of when thinking of models as sets (of something + additional structure)?
Provided:
I can think of models of any ...

**17**

votes

**8**answers

1k views

### The Importance of ZF

It seems as though many consider ZF to be the foundational set of axioms for all of mathematics (or at least, a crucial part of the foundations); when a theorem is found to be independent of ZF, it's ...