So I need to settle an argument a friend of mine is saying you

You can't prove its in contradiction with the system when you don't even really know the rules. Sounds like the guy hasn't even told you how s^2 would work. You could certainly just say s^2 = s. Its not like s behaves like a real number anyways.

Also the reason people don't make dumb things like this is because its basically just the extended real number system that is not a field.

>>8667078
s+1/s=infinity
s*infinity=1+s^2
s^2-s*infinity+1=0
s=(infinity+/-sqrt(infinity^2-4))/2
s=(infinity+/-sqrt(infinity^2))/2
s=(infinity+/-infinity)/2
s=0/2=0 or infinity/2=infinity.
s can't be 0 because you defined it as nonzero. s can't be infinity because infinity is not smaller than a for all a. Both cases are contradictions, therefore s cannot exist.

>>8667112
thanks m8

>>8667116
oh shit actually infinity-infinity isn't 0 it's can be anything so in that case s could exist.

Let s be the smallest positive value in this supposed field of extended reals.

Then s/2 is positive and in the field, and is smaller than s.

Since s, by definition, is the smallest value in the field, but there is a value s/2 that is smaller, there is a contradiction.

Either there can not be a smallest positive value, or the extended reals are not a field, i.e. they are useless.

>>8667166
Correction:
It can't be a field with similar multiplication to the reals.

Thinking it a bit further, s is obviously and trivially defined for finite fields.

>>8667078
(1+s)/s wouldn't equal infinity though

>>8667078
It seems to me that quantified distances should include their inherent reaching towards the next quantifiable but different distance.
It gives numbers "life," that they are in constant change and movement.

s is useless if it's sign is indeterminate (you can't have one constant represent two numbers).

Assume s is positive, then the arithmetic mean of s and 0 is positive, and the arithmetic mean of two numbers is always less than or equal to the largest number.

But s is the smallest positive number, so we have:

(0+s)/2 <= s
s/2 <= s
s/2 = s (by definition of s)
s = 2s
s(1/s) = 2s(1/s) (this is allowed since 1/s is not infinite)
1=2

A contradiction.

A similar proof follows if you assume s is negative.

>>8667166
s/2 may not be smaller than s if it is equal. But this implies s=0.

https://en.wikipedia.org/wiki/Hyperreal_number

>>8667078
Sounds a bit like non-standard analysis.

>>8667078
There is acually a theory of infinitessimals, called "non-standard analysis". It was developed in the 1960s following work by Abraham Robinson at Princeton.

So it is possible, though fiendishly difficult.

https://en.wikipedia.org/wiki/Non-standard_analysis

I'm going to comment, but first of all this whole thing is fucking stupid because it cannot hold when a=s.

That being said, I think there's also a problem with the assumption (or definition?) that 1/s is "non-infinite". How do you define "non-infinite"? I would guess it means the same as "finite", which I would interpret to mean that there exists some other number strictly larger than 1/s, say x > 1/s. But then 1/x < s, contradicting the first part.

>>8667112
this is a beautiful retarded answer to a retarded question

>>8667112
what >>8668124 said
magnifique

Let a=1 and s= 1/2. Clearly, 1/2 must be the smallest number in accordance with the inequalities you've proposed.

OP's friend must therefore be an unparalleled retard.

QED