forcing | Asaf Karagila

Posts tagged forcing

I trained neural nets to do forcing and symmetric extensions

Jun 01 2023, 14:24

So, I spent the last year training a bunch of neural nets how to do forcing, with and without choice, how to work with symmetric extensions, and how to force over symmetric extensions. It was pretty damn good.

If it wasn't clear already, I gave a year-long course here in Leeds on these topics. You can find the notes on the Papers page.

Countable sets of reals

Apr 06 2020, 13:15

One of the classic results of Sierpinski is that if there are as many countable sets of reals as there are reals, then there is a set which is not Lebesgue measurable. (You can find a wonderful discussion on MathOverflow.)

This is fact is used in the paradoxical decomposition theorems (which I often enjoy bringing up as a counter-argument to bad arguments that the Banach–Tarski paradox implies we need to accept that all sets are measurable as an axiom):

Definable Models Without Choice

Jun 07 2018, 23:48

Suppose that a parameter formula defines an inner model. Does that inner model satisfy choice?

Well, obviously, if choice failed then the answer is no, just by taking \(x=x\). But what if we remove that option. Namely, if the inner model is not the entire universe, then choice holds.

Cofinality and the axiom of choice

Nov 29 2015, 22:35

What is cofinality of a[n infinite] cardinal? If we think about the cardinals as ordinals, as we should in the case the axiom of choice holds, then the cofinality of a cardinal is just the smallest cardinality of an unbounded set. It can be thought of as the least ordinal from which there is an unbounded function into our cardinal. Or it could be thought as the smallest cardinality of a partition whose parts are all "small".

Not assuming the axiom of choice the definition of cofinality remains the same, if we restrict ourselves to ordinals and \(\aleph\) numbers. But why should we? There is a rich world out there, new colors that were not on the choice-y rainbow from before. So anything which is inherently based on the ordering properties of the ordinals should not be considered as the definition of an ordinal. So first let's recall the two ways we can order cardinals without choice.

Much needed terminology, that isn't going to happen any time soon

Feb 02 2015, 20:29

One of the reasons I love set theory so much, and specifically choice related research, is that this is an extremely fertile ground for amusing terminology. We have forcing, cardinals, collapsing, we have all sort of gems and rodents at our disposal... we even have a swamp thing.

Here are a few terminological ideas that I doubt are going to be developed by anyone. But if you plan on doing something similar (or if my terminology inspires some proof) feel free to use these terms, and please let me know!

My love-hate relationship with forcing

Sep 10 2014, 08:27

Forcing is great. Forcing is an amazing method. If you can think about it, then you can probably force to make it happen. All it requires is some creativity and rudimentary understanding of the objects that you are working with.

Forcing is horrible. If you can think about it, you can encode it into generic objects. If you can't think about it, you can encode it into generic objects. If you think that you can't encode it into generic objects, then you are probably wrong, and you can still encode it into generic objects.

No uniform ultrafilters

Jul 24 2014, 11:48

Earlier this morning I received an email question from Yair Hayut. Is it consistent without the axiom of choice, of course, that there are free ultrafilters on the natural numbers but none on the real numbers?

Well, of course that the answer is negative. If \(\cal U\) is a free ultrafilter on \(\omega\) then \(\{X\subseteq\mathcal P(\omega)\mid X\cap\omega\in\cal U\}\) is a free ultrafilter on \(\mathcal P(\omega)\). But that doesn't mean that the question should be trivialized. What Yair asked was actually slightly subtler than that: is it consistent that there are free ultrafilters on \(\omega\), but no uniform ultrafilters on the real numbers?

Forcing. This Has To Stop.

Jun 07 2014, 03:36

Most, if not all, set theorists at one point or another were asked by a fellow mathematician to explain how forcing works. And many chose to give as an opening analogy field extensions. You can talk about how the construction of an algebraic closure is a bit similar, since the generic filter is a bit like the maximal ideal you use to make this construction; or you can talk about adding a transcendental number and the things that change as you add it.

But both these analogies would be wrong. They only take you so far, and not further. And if you wish to give a proper explanation to your listener, there will be no escape from the eventual logic and set theory of it all. I stopped, or at least I'm doing my best, using these analogies. I do, however, use the analogy of "How many roots does \(x^{42}-2\) has?" as an example for everyday independence (none in \(\mathbb Q\), two in \(\mathbb R\) and many in \(\mathbb C\)). But this is to motivate a different part of the explanation: the use of models of set theory (e.g. "How can you add a real number??", well how can you add a root to a polynomial?) and the fact that we don't consider the universe per se. Of course, in a model of \(\ZFC\) we can always construct the rest of mathematics internally, but this is not the issue now. Just like we have a model of one theory, we can have a model for another.

Strong chain conditions and preservation of choice principles

Jun 28 2013, 04:04

I recently returned from a wonderful week in Italy, where I attended the Young Set Theorist 2013 conference. I met a lot of new people, some old acquaintances, baffled people with oversized pickles, and most importantly shared and learned some great ideas.

One of the nicer things I'd done was to work with Thomas Johnstone on some preservation theorem related to forcing and choice principles (see also this announcement by Victoria Gitman). In order to clean up a bit the proof, I'll introduce a new definition which is going to slightly extend the ideas originally discussed in Italy. So without further jibber jabber, let's talk mathematics.