Timeline for Why should we care about "higher infinities" outside of set theory?
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37 events
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| Aug 30, 2015 at 16:40 | comment | added | Igor Khavkine | Since no-one else seems to have pointed to it, here's an older question (MO44705) that asked for examples of higher uncountable cardinals naturally occurring outside set theory proper. | |
| Aug 29, 2015 at 13:15 | answer | added | Thomas Benjamin | timeline score: 0 | |
| Aug 28, 2015 at 21:07 | answer | added | Thomas Benjamin | timeline score: 9 | |
| Aug 28, 2015 at 12:34 | answer | added | Avshalom | timeline score: 6 | |
| Aug 28, 2015 at 7:29 | comment | added | Thomas Benjamin | @Cosmonut: Please be so kind as to define what you mean by 'major' in your phrase "major theorems in analysis, algebra, or geometry. I would think if there were, say, a 'classical' statement of analysis that could only be proven in $ZFC+$"there exists a measurable cardinal", would you not deem that result to be "major"? | |
| Aug 27, 2015 at 23:08 | vote | accept | Cosmonut | ||
| Aug 27, 2015 at 11:14 | comment | added | Alexis Hazell | @Asaf: Yes, i was going to link to the Wikipedia entry Ultrafinitism in my comment, but left it out since the Wikipedia entry on Finitism included such a link itself. :-) | |
| Aug 27, 2015 at 10:28 | comment | added | Asaf Karagila♦ | @Alexis: I know a few actual finitists, and they generally hold the belief that there are infinitely many natural numbers, but there is no such thing as "the set of natural numbers". In some sense, they work in ZFC where the axiom of infinity is replaced by its negation. What you're alluding to is probably described as Ultrafinitism. | |
| Aug 27, 2015 at 9:42 | comment | added | Alexis Hazell | "If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. :)" Not necessarily. :-) There are philosophical differences on this point; cf. en.wikipedia.org/wiki/Finitism. | |
| Aug 27, 2015 at 9:15 | comment | added | Joel David Hamkins | @TerryTao It may be interesting to mention that in the separable case, the Hahn-Banach theorem is equivalent in reverse mathematics to $\text{WKL}_0$, a comparatively weak theory. See en.wikipedia.org/wiki/…. | |
| Aug 27, 2015 at 8:52 | answer | added | Thomas Klimpel | timeline score: 2 | |
| Aug 27, 2015 at 7:38 | comment | added | Andreas Blass | I don't understand the statement "... suppose we did not have the PowerSet Axiom, One could still prove that the reals had a higher cardinality than the naturals ...." Without the power set axiom, you don't know that there is a set of all reals. Are you working in a theory with proper classes so that you can still talk about the cardinality of the class of reals? | |
| Aug 27, 2015 at 7:22 | comment | added | David Roberts♦ | You may also be interested in predicative mathematics: ncatlab.org/nlab/show/predicative+mathematics. It doesn't get rid of higher cardinalities, insomuch as this still makes sense, but does reject the notion of power set (still keeping, for instance, function sets more generally, but rejecting a subobject classfying set, which is classical mathematics is {0,1}) | |
| Aug 27, 2015 at 7:22 | answer | added | Joel David Hamkins | timeline score: 30 | |
| Aug 27, 2015 at 4:01 | comment | added | Cosmonut | Terry: "To me, it's not so much that the multiplicity of uncountable cardinals comes up directly all that much in mathematics, but rather that they are inevitable byproducts of the very useful set theory axioms" Thanks ! That's very helpful. | |
| Aug 27, 2015 at 3:46 | comment | added | Terry Tao | Perhaps an analogy will help. Your question seems to me to be similar to "why should we care about $0.999\dots = 1$ outside of analysis?". On the one hand, it would be rare to have a situation outside of analysis in which one could start with the assumption $0.999\dots \neq 1$ and accidentally end up with a more blatant contradiction. On the other hand, the effort required to purge $0.999\dots = 1$ from being a consequence of the foundations of mathematics outside of analysis, would greatly reduce the ability to efficiently work on those areas for very little gain. | |
| Aug 27, 2015 at 3:31 | comment | added | Terry Tao | I think it actually quite hard to fence off the Power Set Axiom (and a fortiori, the existence of different uncountable cardinalities) from the rest of mathematics. For instance, in analysis, establishing the Hahn-Banach theorem or the existence of the Stone-Cech compactification requires pretty much all of the axioms of set theory. To me, it's not so much that the multiplicity of uncountable cardinals comes up directly all that much in mathematics, but rather that they are inevitable byproducts of the very useful set theory axioms that we rely quite heavily in mathematics. | |
| Aug 27, 2015 at 3:26 | comment | added | Cosmonut | Terry, I was coming at this from a different angle. For instance, suppose we did not have the Power Set Axiom. One could still prove that the reals had a higher cardinality than the naturals as in the link below for example. boolesrings.org/scoskey/my-favorite-proof-that-r-is-uncountable Would there be any theorems outside of set theory - hence, not relying on the Power Set Axiom, perhaps - that would force us to conclude that some mathematical object had a cardinality higher than R ? | |
| Aug 27, 2015 at 3:19 | comment | added | Burak | What is the "fastest" way of proving that there is a Lebesgue measurable non-Borel set? This fact can be proven without knowing that there are more Lebesgue measurable sets than Borel sets (for example you can construct a non-Borel analytic set and then prove that analytic sets are Lebesgue measurable). However, I cannot think of a simpler proof than a cardinality argument. | |
| Aug 27, 2015 at 3:16 | comment | added | Cosmonut | Burak, would you please elaborate on your comment about different uncountable cardinals ? Are there any major theorems outside Set Theory which crucially depend on having different uncountable cardinals ? | |
| Aug 27, 2015 at 3:04 | comment | added | Terry Tao | Given that Cantor's theorem is more or less immediate from the Power Set Axiom together with relatively uncontroversial axioms, perhaps the real question here is "why should we admit the Power Set Axiom in mathematics outside of set theory?" | |
| Aug 27, 2015 at 2:44 | comment | added | Burak | I do not understand the question. There are three types of cardinalities: finite, countably infinite and uncountably infinite (simply because this is a trichotomy). So you cannot possibly "conflict" with any major theorems. I guess you want to ask why we should care about different uncountable cardinals, and there are way too many reasons for that. | |
| Aug 27, 2015 at 0:54 | comment | added | David Roberts♦ | You may be interested in a foundation where one can form each of the finitely iterated power sets $P^n(\mathbb{N})$, but draw no conclusion about anything bigger. Then one is assured of having spaces of functions that turn up in analysis etc, starting from existing sets. See: arxiv.org/abs/1212.6543, and this question, phrased in terms of higher cardinalities arising from Replacement: mathoverflow.net/questions/208711/who-needs-replacement-anyway | |
| Aug 27, 2015 at 0:53 | comment | added | Monroe Eskew | You could just work in third order arithmetic. | |
| S Aug 27, 2015 at 0:48 | history | suggested | user5794 | CC BY-SA 3.0 |
Less antagonistic title
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| Aug 27, 2015 at 0:44 | comment | added | user5794 | @YemonChoi: Isn't the question identifying the cardinality of all non-countable infinite sets into an equivalence class called "uncountable"? | |
| Aug 27, 2015 at 0:41 | comment | added | Yemon Choi | A question from the OP: does "conflict with" mean something different from "lead to conclusions inconsistent with"? | |
| Aug 27, 2015 at 0:39 | comment | added | Yemon Choi | @trb456 Your question seems different from the original, since you ask "if we only know" rather than "we assert that two sets have the same cardinality" | |
| Aug 27, 2015 at 0:38 | review | Suggested edits | |||
| S Aug 27, 2015 at 0:48 | |||||
| Aug 27, 2015 at 0:35 | comment | added | user5794 | @CarlMummert: Sure, I'll ask that question: if we only know that the Stone-Cech compactification of the naturals is not countable, what do we lose in terms of major theorems outside of set theory? | |
| Aug 27, 2015 at 0:35 | comment | added | Asaf Karagila♦ | Can you phrase the title in a less antagonizing fashion? | |
| Aug 27, 2015 at 0:25 | comment | added | Carl Mummert | I am not sure what you are asking. Surely there are common objects that have cardinality larger than the reals - the Stone-Cech compactification of the naturals, $\beta\mathbb{N}$, is a key example with cardinality $2^\mathfrak{c}$. Are you asking whether it would be possible for someone to put their head in the sand and never ask what the cardinality that set has? | |
| Aug 27, 2015 at 0:20 | comment | added | Francis Adams | This type of set theory actually has a name, Pocket Set Theory. en.wikipedia.org/wiki/Pocket_set_theory | |
| Aug 27, 2015 at 0:12 | comment | added | Steven Stadnicki | Fixed-point theorems seem like a likely candidate for serious problems, too... | |
| Aug 27, 2015 at 0:11 | comment | added | David Roberts♦ | What's the cardinality of the space of operators on a separable Banach space? | |
| Aug 27, 2015 at 0:06 | review | First posts | |||
| Aug 27, 2015 at 0:25 | |||||
| Aug 26, 2015 at 23:54 | history | asked | Cosmonut | CC BY-SA 3.0 |