The Continuum Hypothesis (was: Does Hubble's Law tell us...)

[John Clark]

On Fri, Aug 29, 2025 at 3:09 PM Alan Grayson <agrays...@gmail.com> wrote:

>>> Actually, sometimes even in pure mathematics we can't always reach absolute conclusions, a good example of which is the CONTINUUM HYPOTHESIS. AG 

>> But it has been proven you can assume  that the continuum hypothesis is true or you can assume that the continuum hypothesis is not true, but neither assumption will produce a contradiction to existing mathematics. It doesn't matter, so to my mind that indicates that the continuum hypothesis is just not very important. 

> What's "important" here is in the mind of mathematicans. And IMO you've misstated the result. AG

In 1940 Kurt Gödel proved that the truth of the Continuum Hypothesis is consistent with existing mathematics, that is to say if it's true then it would not change anything. In 1963 Paul Cohen proved that the NEGATION of the Continuum Hypothesis is ALSO consistent with existing mathematics. As a result of these developments I don't think the Continuum Hypothesis is meaningless but I do think it's unimportant. I say that because, if neither the truth nor the falsehood of a conjecture would change anything and if the word has any meaning then that conjecture is "unimportant". 

And we can't just add the Continuum Hypothesis as an axiom because an axiom needs to be simple and self-evidently true, and the Continuum Hypothesis is neither of those things. And the same thing could be said about its negation.

 John K Clark    See what's on my new list at  Extropolis

aoc

[Giulio Prisco]

Doesn’t seem unimportant to me, quite the contrary. I guess we haven’t found yet a good set of axioms.

 John K Clark    See what's on my new list at  Extropolis

aoc

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[Alan Grayson]

On Saturday, August 30, 2025 at 4:04:00 AM UTC-6 John Clark wrote:

On Fri, Aug 29, 2025 at 3:09 PM Alan Grayson <agrays...@gmail.com> wrote:

>>> Actually, sometimes even in pure mathematics we can't always reach absolute conclusions, a good example of which is the CONTINUUM HYPOTHESIS. AG 

>> But it has been proven you can assume  that the continuum hypothesis is true or you can assume that the continuum hypothesis is not true, but neither assumption will produce a contradiction to existing mathematics. It doesn't matter, so to my mind that indicates that the continuum hypothesis is just not very important. 

> What's "important" here is in the mind of mathematicans. And IMO you've misstated the result. AG

In 1940 Kurt Gödel proved that the truth of the Continuum Hypothesis is consistent with existing mathematics, that is to say if it's true then it would not change anything. In 1963 Paul Cohen proved that the NEGATION of the Continuum Hypothesis is ALSO consistent with existing mathematics. As a result of these developments I don't think the Continuum Hypothesis is meaningless but I do think it's unimportant. I say that because, if neither the truth nor the falsehood of a conjecture would change anything and if the word has any meaning then that conjecture is "unimportant". 

It's only important if we want to understand the reality of transfinite cardinal numbers, and the conclusion that there is no infinite set strictly between the integers and the real numbers. AG 

[Alan Grayson]

On Saturday, August 30, 2025 at 4:17:08 AM UTC-6 Alan Grayson wrote:

On Saturday, August 30, 2025 at 4:04:00 AM UTC-6 John Clark wrote:

On Fri, Aug 29, 2025 at 3:09 PM Alan Grayson <agrays...@gmail.com> wrote:

>>> Actually, sometimes even in pure mathematics we can't always reach absolute conclusions, a good example of which is the CONTINUUM HYPOTHESIS. AG 

>> But it has been proven you can assume  that the continuum hypothesis is true or you can assume that the continuum hypothesis is not true, but neither assumption will produce a contradiction to existing mathematics. It doesn't matter, so to my mind that indicates that the continuum hypothesis is just not very important. 

> What's "important" here is in the mind of mathematicans. And IMO you've misstated the result. AG

In 1940 Kurt Gödel proved that the truth of the Continuum Hypothesis is consistent with existing mathematics, that is to say if it's true then it would not change anything. In 1963 Paul Cohen proved that the NEGATION of the Continuum Hypothesis is ALSO consistent with existing mathematics. As a result of these developments I don't think the Continuum Hypothesis is meaningless but I do think it's unimportant. I say that because, if neither the truth nor the falsehood of a conjecture would change anything and if the word has any meaning then that conjecture is "unimportant". 

It's only important if we want to understand the reality of transfinite cardinal numbers, and the conclusion that there is no infinite set strictly between the integers and the real numbers. AG 

But of course, that's what we don't know! AG 

[Brent Meeker]

On 8/30/2025 3:03 AM, John Clark wrote:

On Fri, Aug 29, 2025 at 3:09 PM Alan Grayson <agrays...@gmail.com> wrote:

>>> Actually, sometimes even in pure mathematics we can't always reach absolute conclusions, a good example of which is the CONTINUUM HYPOTHESIS. AG 

>> But it has been proven you can assume  that the continuum hypothesis is true or you can assume that the continuum hypothesis is not true, but neither assumption will produce a contradiction to existing mathematics. It doesn't matter, so to my mind that indicates that the continuum hypothesis is just not very important. 

> What's "important" here is in the mind of mathematicans. And IMO you've misstated the result. AG

In 1940 Kurt Gödel proved that the truth of the Continuum Hypothesis is consistent with existing mathematics, that is to say if it's true then it would not change anything.

 In 1963 Paul Cohen proved that the NEGATION of the Continuum Hypothesis is ALSO consistent with existing mathematics. As a result of these developments I don't think the Continuum Hypothesis is meaningless but I do think it's unimportant. 

It may be unimportant, I don't know of anything it's led to, but you're wrong to say that adding it as an axiom to would not change anything.  It would not change anything already proven, but it might make some theorems, yet unformulated, decidable which were not before.

I say that because, if neither the truth nor the falsehood of a conjecture would change anything and if the word has any meaning then that conjecture is "unimportant". 

And we can't just add the Continuum Hypothesis as an axiom because an axiom needs to be simple and self-evidently true, and the Continuum Hypothesis is neither of those things. And the same thing could be said about its negation.

Where did you get that from, and who decides what is simple and self-evident?

Brent