Some real miscellany for today!:
1) start with some actual mathy-ness:
The continuum hypothesis involves a well-known conundrum (considered undecidable without new set theory) among mathematicians over whether any infinite sets exist between aleph-zero and aleph-one. But I don't recall hearing any arguments over whether there could be infinite sets between any other alephs (say, aleph-three and aleph-four) — of course these higher sets are all power sets, but I can’t recall ever reading any “proof” that there can be no set in-between… I assume this is a long-settled simple question, but am not sure what the simple answer is! or is it somehow axiomatic with no real proof... or does it not even much matter since there's already an infinity of infinite sets???
ADDENDUM: I've now posed this question to a local retired math professor/number-theorist and he didn't know the answer, so at least I feel better at this point that it may not be a simple or dumb question to ask! (but have not heard from anyone else)
ADDENDUM: I've now posed this question to a local retired math professor/number-theorist and he didn't know the answer, so at least I feel better at this point that it may not be a simple or dumb question to ask! (but have not heard from anyone else)
2) Next, no math, just a question that I asked on Twitter but got little response to, so will try here:
Google search has given me sporadically crappy results for several weeks (sometimes NO results, and sometimes results having no or little bearing on what I’m searching for), so I’ve switched over to Bing for now, but just wondering what search engines other math-types are happy with (it’s not a privacy issue or any other concern, strictly quality/relevance of results)? And are others experiencing issues with Google search — seems like some real glitch involved?
3) Finally, just a note… Two of the the greatest television shows ever when I was younger were Carl Sagan’s “Cosmos” and Jacob Bronowski’s “The Ascent of Man.” I’ve sometimes used Bronowski quotes and videos here on the blog, and just as “Cosmos” was re-done a few years back I feel like “Ascent of Man” should be re-run or re-done for each new generation.
Anyway, I stumbled across Bronowski on the Web a few days back and suddenly realized that he was trained as a mathematician (somehow I had him pegged in my mind as a physical scientist). Also didn’t realize he had died at the age of 66 (much younger than I thought) just one year after “Ascent…” was completed in 1973. Additionally discovered that “Ascent…” was originally commissioned by Sir David Attenborough. Just all interesting tidbits to me… and here is Dr. Bronowski voicing some of his thoughts about mathematics:
3 comments:
I think you've got the wrong end of the stick, concerning the continuum hypothesis. Aleph-one is, by definition, the next cardinality after aleph-zero. The continuum hypothesis is there are no sets of cardinality strictly between that of the integers (that's aleph-zero) and that of the reals. In other words, the continuum hypothesis is that the cardinality of the reals is aleph-one.
I see you mention power sets. The cardinality of the reals is the same as the cardinality of the power set of the integers, so another way to state the continuum hypothesis is that there is no set of cardinality strictly greater than that of the integers but strictly less than that of the set of all sets of integers.
The generalized continuum hypothesis says not only is there no cardianlity strictly between the integers and the power set of the integers, there's also no cardinality between the power set of the integers and the power set of the power set of the integers, nor between the power set of the power set and the power set of the power set of the power set, and so on. This has also been proved to be undecidable within the usual axioms of set theory.
OK, thanks Gerry; I'm accustomed to seeing the Continuum Hyp. referring to the gap between the integers and the reals (or the power set of integers), but just didn't recall a "generalized" form that references the gap between any two power sets. I guess I wonder if the "proof" of undecidability is identical in both cases or slightly different, since the integers are not a power set, so you're comparing a non-power set to a power set in one case, but 2 successive power sets in the other?
Is it fair to say (due to undecidability) that all the alephs above zero are essentially numbered by fiat, and if we used a different set theory what we call aleph-four, for example, might turn out to be say aleph-eight?
I regret that I am not familiar with the details of the undecidability proofs, and so can't tell you how or whether the undecidability proof for the Generalized Continuum Hypothesis (GCH) differs from that for the Continuum Hypothesis (CH). I'm not sure I understand what you're getting at in your last paragraph, but maybe it's something like this: it is consistent with the other axioms to take the cardinality of the reals to be aleph-one (that is, CH is consistent with the other axioms), but it is also consistent with the other axioms to take the cardinality of the reals to be aleph-two (so there would be exactly one cardinality strictly between the integers and the reals), or aleph-three, or aleph-four....
The set theorists (and I don't qualify for membership there) use beth-one, beth-two, and so on, for the cardinalities of the successive power sets. Beth-zero = aleph-zero, and after that beth-(n + 1) = 2^(beth-n). GCH says beth-n = aleph-n for all n. But if you reject GCH, then it's consistent to have, say, beth-4 = aleph-8.
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