The cool thing about the congruence tower problem is that it is downwards absolute under forcing; if there is a frame L in a forcing extension V[G] whose congruence tower terminates at an ordinal alpha, then there is a frame in M in V whose congruence tower also terminates at the same ordinal alpha.

]]>Another point is that the $2k$-complement can be replaced by any involution (even case) or any involution with exactly one fixed point (odd case).

]]>“Although the arguments here show that $KM$ is strictly stronger than $ZFC$ in consistency strength, it is not really very much stronger…”, especially that “if $\kappa$ is an inaccessible cardinal, then it is not difficult to argue in $ZFC$ that $$ is a model of $KM$”. If one assumes $\kappa$ is the least inaccessible cardinal so that $V_{\kappa}$ is a model of ZFC$ + “There is no inaccessible cardinal.” then by the usual understanding of $V_{|kappa+ 1}$ (i.e. $V_{\kappa +1}$ = $Powerset$($V_{\kappa}$) since one is arguing in $ZFC$), both $\kappa$ and $V_{\kappa}$ are sets in $V_{\kappa + 1}$ and the model of $KM$ in question is not merely a model of $KM$ but is a model of $KM$ + “There is a proper class of inaccessible cardinals” (at least). Am I correctly understanding the state of affairs and if not, would you please point out the errors in my under standing? Thanks. By the way, it is a very interesting blogpost.

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Accepted, March 2018.]]>

Accepted, November 2017.]]>

Accepted November 2017.]]>

The book by Hindman&Strauss has a section on filters and compactifications in a later chapter — that might be suitable for 2).

]]>Possibly relevant is my October 2004 sci.math post “Generalized Quantifiers” (URLs below). FYI, the Math Forum version has a lot of strange formatting errors. See also Brian Thomson’s 1985 book “Real Functions”, and see Thomson’s earlier 2-part survey Derivation bases on the real line (which contain examples and side-detours not in his book).

google sci.math URL:

https://groups.google.com/forum/#!msg/sci.math/rhZEhXynVLQ/MI0MJ0ZQIvoJ

Math Forum sci.math URL:

http://mathforum.org/kb/message.jspa?messageID=3556191

Regarding Moving around every few years for postdocs would be exhausting

, I wholeheartedly agree! Every such decision has pros and cons. They are usually not obvious at the outset. Moreover, they change over time. The lesson I learned though my own course is that every decision is right, at least at the time you make it, and there is never any point regretting it later on… Just keep on doing what you do best all the time!

We went on to look at the paradoxical situations that arise when someone gets a positive test result for a rare disease. Should they be worried? If the test is 99% accurate, but the disease occurs in say, 1 in million, then a positive result is not so worrisome: in million people, there will be about 1 true positive result, and about 10,000 false positives, since 1% of a million is 10,000. So the odds that you’ve actually got the disease, given that you tested positive, is 1 in 10,000.

But your situation is completely different! It would be as though we had calculated the odds of getting HHHTTT, and then when I actually flipped the coin on stage, i actually got the same pattern HHHTTT. Totally weird! And very unlikely. But you know, if it wasn’t that, it would have been some other totally unlikely thing, like getting all green lights, or all red lights, or getting the serial number 123456 on your receipt at Starbucks.

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