# Do mathematicians assume certain mathematical hypothesis to be true while waiting for a proof? **URL:** \ **Category:** Factual Questions\ **Created:** [May 20, 2024, 8:08pm UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824 "2024-05-20T20:08:56Z")\ **Posts on this page:** 12\ **Page:** 3
**Author:** ![EnolaStraight](https://avatars.discourse-cdn.com/v4/letter/e/41988e/32.png) [@EnolaStraight](https://boards.straightdope.com/u/EnolaStraight)\ **Post date:** [June 2, 2024, 1:32pm UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/41 "2024-06-02T13:32:07Z")
What if you state that the very fundamentals of chemistry, physics, and mathematics are in fact based on entirely different principles…like 1x1=2 ? 🤔 [Unlocking the Universe: Exploring Terrence Howard’s Revolutionary Theories – The Personal Site of Lalo Morales (laloadrianmorales.com)](https://laloadrianmorales.com/blog/unlocking-the-universe-exploring-terrence-howards-revolutionary-theories/) ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [June 2, 2024, 5:35pm UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/42 "2024-06-02T17:35:58Z")
For me the AC is incredibly, “well duh” obvious. At first take. Then I start considering some situations I would want to use it. Then it becomes, “Well, maybe not.” And I started really digging into it and I end up “This is one really big, unfounded premise.” As opposed to CH where Cantor should have just declared Aleph-1 to be the cardinality of the Reals and moved on. Left the “in between” question to the mega basement dwelling Math nerds. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 2, 2024, 6:02pm UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/43 "2024-06-02T18:02:56Z")
> [@ftg](#): > > As opposed to CH where Cantor should have just declared Aleph-1 to be the cardinality of the Reals and moved on. Left the “in between” question to the mega basement dwelling Math nerds. Is it even proven that there is an Aleph-1? That is to say, a smallest cardinal that’s strictly larger than Aleph-0? Maybe, not only are there cardinals between Aleph-0 and 2^Aleph-0, but there are are an infinite number of them, and they’re not well-ordered. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 2, 2024, 7:42pm UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/44 "2024-06-02T19:42:20Z")
I have to think about it more rigorously, but I think (without resorting to any axiom of choice) you can define Aleph-1 as the cardinality of the least ordinal such that there is no injective map from it into Aleph-0. It will really be a successor cardinal, that is, a minimal cardinal greater than Aleph-0. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [June 2, 2024, 10:36pm UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/45 "2024-06-02T22:36:30Z")
But without AC, I don’t see why there should be such a least ordinal. I am so used to AC that I find it hard to think of set theory without it (although I know many models of ZF that lack AC). Mega bottom dwelling math nerd ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 2, 2024, 10:50pm UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/46 "2024-06-02T22:50:59Z")
Any ordinal, and the class of all ordinals, is well-ordered with respect to set membership… I think that does not depend on the axiom of choice? Without AC one does need to be careful though. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 3, 2024, 12:38am UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/47 "2024-06-03T00:38:02Z")
The key here seems to be [Hartogs’s Lemma](https://en.wikipedia.org/wiki/Hartogs_number): if X is any set, then \{\,\alpha\in\mathrm{Ord} \mid \alpha \le X\,\} is a set, [where A \le B here means there exists an injection of A into B], which is an ordinal, and is the minimal ordinal with no injection into X. As a corollary, if \kappa is a well-ordered cardinal, we can define its successor \kappa^+ as \{\,\alpha\in\mathrm{Ord} \mid \alpha\le\kappa\,\}; then \kappa^+ is an ordinal, \kappa^+ is a cardinal, \kappa\<\kappa^+, and \kappa^+\not\le\kappa (and there is no smaller ordinal with those properties!). Point is, even without AC it makes sense to define \aleph\_1 as “the smallest uncountable cardinal”, and there is nothing strictly in-between \aleph\_0 and \aleph\_1. ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [June 9, 2024, 1:14am UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/48 "2024-06-09T01:14:41Z")
Y’all deserve this oldie but baddie: Q. What’s yellow and equivalent to the Axiom of Choice? A. Zorn’s Lemon. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [June 10, 2024, 12:57am UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/49 "2024-06-10T00:57:44Z")
What’s purple and commutes? An Abelian grape ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [June 10, 2024, 6:07pm UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/50 "2024-06-10T18:07:32Z")
I’ve heard he’s working from home now, though. ---
**Author:** ![griffin1977](https://avatars.discourse-cdn.com/v4/letter/g/977dab/32.png) [@griffin1977](https://boards.straightdope.com/u/griffin1977)\ **Post date:** [June 11, 2024, 1:06pm UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/51 "2024-06-11T13:06:04Z")
So I just watched a [YouTube video](https://youtu.be/kmAc1nDizu0?si=m35rrI0nnpD_VIQY) that had a cool example of this. They have come up with a Turing Machine that only halts if and only if the Reimann Hypothesis is true: [Quanta Magazine](https://www.quantamagazine.org/how-the-slowest-computer-programs-illuminate-maths-fundamental-limits-20201210/#:~:text=They%20identified%20a%20744%2Drule,Millennium%20Problems%E2%80%9D%20worth%20%241%20million). ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 12, 2024, 12:05pm UTC](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824/52 "2024-06-12T12:05:48Z")
A great many unsolved mathematical problems can be reduced to halting problems. Which is a good intuitive demonstration of the fact that the halting problem is very difficult (of course, “known to be impossible” is a special case of “very difficult”). [Previous page](https://boards.straightdope.com/t/do-mathematicians-assume-certain-mathematical-hypothesis-to-be-true-while-waiting-for-a-proof/1001824.md?page=2)