Problem with Cantors Diagonalization Proof.

talanum1 said:
I can be wrong, and I'll admit it if I'm wrong,
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talanum1 said:
but in this case I'm not wrong.
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talanum1 said:
But now you must continue the list and you can't till infinity because you would have to look at the digit "at" infinity and there is no such digit.

I'm not treating infinity like a number.
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Same rule applies regardless. If it were theoretically possible to examine the infinite digit, it would still be somewhere from 0 to 9 and would still be replaced by a choice of either 7 or 8 alternatives. Same as any preceding digit. So no difference anyway.
 
Bliksis said:
The aliens will know.
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Nah, these aliens must have 17 fingers or something, they're only good at particle physics (sometimes)

images
 
Brian_G said:
Same rule applies regardless. If it were theoretically possible to examine the infinite digit, it would still be somewhere from 0 to 9 and would still be replaced by a choice of either 7 or 8 alternatives. Same as any preceding digit. So no difference anyway.
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OK, I see now that I am wrong: the argument just requires the specification of the n'th digit's change and one need not know what the digit "at" infinity changes to, just that it changes. But there still is doubt that one can take the digit "at" infinity and change it.

One needs an axiom to this effect and then derive the consequences. If the consequences are contradictory one must abandon the axiom and with it Cantor's proof.
 
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What would the process looks like if one take the n'th digit and change it an then let n tend to infinity.

Well one time we do the operation the digit changes to say 1, another time it changes to 2. So we have a contradiction so it must be impossible to change the n'th digit and then let n tend to infinity.
 
talanum1 said:
What would the process looks like if one take the n'th digit and change it an then let n tend to infinity.

Well one time we do the operation the digit changes to say 1, another time it changes to 2. So we have a contradiction so it must be impossible to change the n'th digit and then let n tend to infinity.
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You’re still misunderstanding infinity. It’s a process/function/method with no end.
 
talanum1 said:
...But there still is doubt that one can take the digit "at" infinity and change it.

One needs an axiom to this effect and then derive the consequences. If the consequences are contradictory one must abandon the axiom and with it Cantor's proof.
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You're over thinking it, trying to give infinity itself some suspected special concept here, but it's about the forever journey that can actually be fully examined, not about any impossible destination. Your post was fine before this edit.

Maybe give it a couple of days' rest.
 
If it's a process without end one can never take the infinite'th digit and change it.

Best one can do is take the n'th digit for n large enough, but then the process won't agree if done two times.
 
talanum1 said:
That's what I keep on saying.
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You're ok with the irrational real number itself being without digits' end, of typical infinite nature. Yet you can't accept or conceive of a simple enough change to that number, digit by digit, which doesn't change its infinite nature structure at all.

You're stuck in the loop of what you won't let go of, agreeing and then convincing yourself to disagree again.

I'll have to leave it there.
 
You're stating you can change pi digit by digit for all it's digits.
 
talanum1 said:
You're stating you can change pi digit by digit for all it's digits.
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I'm surprised you think so. I'll try once more..

One is just borrowing pi or another number of that mathematical type - as an otherwise unrelated exercise in mathematics only, not to try warp its own established reality if that's what you meant.

As I understand it the result is merely a phenomenon of interest, like many quantum ideas, not an effort at a solid theory like the nature of circles.
 
Brian_G said:
As I understand it the result is merely a phenomenon of interest, like many quantum ideas, not an effort at a solid theory like the nature of circles.
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To me it seems as concrete as anything.

You sidestepped my question.
 
talanum1 said:
To me it seems as concrete as anything.

You sidestepped my question.
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If you honestly think thousands of mathematicians are cooperating in altering pi, then you are ill my man.

But if I've misread that as well, try redefine your question with enough detail so we can all understand,
 
talanum1 said:
You're stating you can change pi digit by digit for all it's digits.
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Let f(i) be the i-th fractional digit of pi. The number x is defined as sum(((f(i)+1) mod 10).10^-i) for i=0 to infinity.

x differs from pi in every digit.
 
Now I'm starting to have bonkers thoughts as well; could the nature of infinity itself somehow alter what's believed obvious "in the end"?

For instance, although we know how to prove it, why exactly does .999...recurring equal 1 ? It never should quite get there, yet it does :unsure:o_O🥸
 
Brian_G said:
Now I'm starting to have bonkers thoughts as well; could the nature of infinity itself somehow alter what's believed obvious "in the end"?

For instance, although we know how to prove it, why exactly does .999...recurring equal 1 ? It never should quite get there, yet it does :unsure:o_O🥸
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There is no real number between 1 and 0.9999… recurring. This is only true for equal numbers.
 
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