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October 3rd, 2019, 07:45 AM   #1
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Comparing large numbers without calculus

A curious logical consequence of the principle of non-contradiction is that a contradiction implies any statement; if a contradiction is accepted as true, any proposition (or its negation) can be proved from it.

Let's try (*)$\displaystyle 66^{77} > 77^{66}$ .

I will prove (*) using sgn function(analytic one) . By this equallity : $\displaystyle sgn(x)=-1+2\lim_{s\rightarrow \infty} (1+e^{-2sx})^{-1}$.
Let $\displaystyle x=66^{77}-77^{66}$ and suppose $\displaystyle 77^{66}>66^{77}$ or $\displaystyle x<0 \equiv sgn(x)<0$.
$\displaystyle sgn(x)=-1+2\lim_{s\rightarrow \infty} \frac{1}{1+e^{-2s\cdot sgn(x)(77^{66}-66^{77})}}\: \equiv \: -1+2\lim_{(s,x(s) )\rightarrow \infty} \frac{1}{1+e^{-2s|x(s)|}}.$
$\displaystyle sgn(x)=-1+2\cdot \frac{1}{1+e^{-2\cdot \infty |\infty| }}=-1+2=1>0$ which is a contradiction of $\displaystyle sgn(x)<0 \;$, therefore (*) is proven .

Last edited by idontknow; October 3rd, 2019 at 07:49 AM.
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October 3rd, 2019, 09:22 AM   #2
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If I understand correctly, the principle of explosion...

https://en.m.wikipedia.org/wiki/Principle_of_explosion

...is the reason for the principle of noncontradiction:

https://en.m.wikipedia.org/wiki/Law_of_noncontradiction

It is true that $66^{77} > 77^{66}$ isn’t it? Is there an issue? I guess I’m confused.
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October 3rd, 2019, 10:21 AM   #3
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are you seriously leaving

$\dfrac{1}{1+e^{-2\infty|\infty|}}$

as a directly evaluable term?

I mean it's pretty clear that it's 1 but you can't just leave it like that.
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October 3rd, 2019, 10:53 AM   #4
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Quote:
Originally Posted by romsek View Post
are you seriously leaving

$\dfrac{1}{1+e^{-2\infty|\infty|}}$

as a directly evaluable term?

I mean it's pretty clear that it's 1 but you can't just leave it like that.
I agree , I evaluated it in short-terms.
In this case $\displaystyle \lim_{a\rightarrow \infty } \lim_{b\rightarrow \infty} e^{-ab} = \lim_{b\rightarrow \infty } \lim_{a\rightarrow \infty} e^{-ab}$ holds true .
The limit is with two variables .

Last edited by idontknow; October 3rd, 2019 at 10:56 AM.
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