Timeline for Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$
Current License: CC BY-SA 4.0
15 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Aug 12, 2018 at 5:14 | answer | added | Mohammad Golshani | timeline score: 7 | |
| Jul 30, 2018 at 3:59 | comment | added | Duchamp Gérard H. E. | You make me doubt of what I was trained to (I am not a logician). You confirm that, in any context, $|\mathbb{R}|=2^{\aleph_0}$ though ? | |
| Jul 30, 2018 at 3:49 | answer | added | Noah Schweber | timeline score: 9 | |
| Jul 30, 2018 at 2:28 | comment | added | Iian Smythe | Justin Moore has an article on this very subject: What makes the continuum $\aleph_2$ | |
| Jul 30, 2018 at 1:37 | history | edited | Morteza Azad | CC BY-SA 4.0 |
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| Jul 30, 2018 at 0:41 | comment | added | Morteza Azad | @NoahSchweber (+1) Interesting! So it is not true to attribute the set-theoretic "common belief", $2^{\aleph_0}=\aleph_2$, to Cohen! Though, in the case of Goedel, I am almost sure that he somewhere expressed such an opinion in favor of the common belief. By the way, thanks for the explanations regarding Woodin's $\Omega$-logic approach. | |
| Jul 30, 2018 at 0:35 | comment | added | Noah Schweber | From the answers to that question we also get Cohen on the value of the continuum. | |
| Jul 30, 2018 at 0:32 | comment | added | Noah Schweber | Following up re: Woodin: from this answer and its attendant comments I believe that the issue was that part of Woodin's "$\Omega$-program" relied on the claim that from the $\Omega$-conjecture it follows that the set of $\Omega$-validities is definable in (or around?) $H(\mathfrak{c}^+)$. This claim, or rather the argument for it given, was flawed, and this was demonstrated in Sargsyan's thesis. Meanwhile, I think the answers over there say a bit about why $\aleph_2$ is attractive, too, especially Justin Moore's. | |
| Jul 30, 2018 at 0:26 | comment | added | Morteza Azad | @NoahSchweber The essential difference is that I am particularly focusing on the assumption $2^{\aleph_0}=\aleph_2$ and how it affects the mathematical machinery and theorems in and out of set theory. The question is not actually about what the true value of continuum could be but about all mathematical evidence which can justify and support one particular choice of the size of the continuum, namely $\aleph_2$, which due to the common consensus in (at least part of) the set theory community is considered the most likely value of $|\mathbb{R}|$. I wonder how justified it is mathematically. | |
| Jul 30, 2018 at 0:18 | comment | added | Noah Schweber | Now that I remember it, I'm not really sure Q2 is sufficiently different from this old question. | |
| Jul 30, 2018 at 0:10 | comment | added | Morteza Azad | @NoahSchweber As for Cohen, I also have heard two different stories (i.e. $|\mathbb{R}|=\aleph_2$ or very large). I am not sure which is historically accurate and that is why I asked for reliable references here! Maybe just like Woodin he also changed his mind at some point! Not quite sure anyway! | |
| Jul 30, 2018 at 0:03 | comment | added | Noah Schweber | Re: Woodin changing his mind, I believe Sargsyan's thesis showed that there was a serious flaw in that approach; maybe someone familiar with inner model theory can comment (or correct me)? Incidentally, my recollection is that Cohen said quite the opposite: that he tended towards the continuum being quite large, e.g. above the first weakly inaccessible. And re: question 2, strong forcing axioms imply $2^{\aleph_0}=\aleph_2$ and give nice properties of the continuum. | |
| Jul 30, 2018 at 0:00 | history | edited | Morteza Azad | CC BY-SA 4.0 |
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| Jul 29, 2018 at 23:48 | history | asked | Morteza Azad | CC BY-SA 4.0 |