/sci/ - Science & Math

The "trick" about this expression is that a [math]1[/math] is obscured, namely

[math]1=\dfrac{a\cdot c+b}{a\cdot c+b}=c\cdot\dfrac{a+b/c}{a\cdot c+b}[/math]

As a consequence

[math]\dfrac{1}{c} = \dfrac{a+b/c}{a\cdot c+b}[/math]

where the right hand side looks like it depends on [math]a[/math] and [math]b[/math] but actually it doesn't.

Use this relation with [math]a=1, b=x, c=\sqrt{1+x^2}[/math]

>>11057242
What's not countable is the number of functions from N to B={0,1}, that's writeen B^N (cardinal exponentiation)

But you can take arbitrarily number of slots list like in a matrix row and put in arbitrarily large numbers (that's written w^w, here no such slot list is not unending in itself, but all such lists are in it). Then you can look at arbitrary long listing of such arbitrarily long matrices w^w^w (this is again all the lists of lists of lists of any (fixed) size), and arbitrarily long of those w^w^w^w, and so on w^w^w^w^... And you can take this to infinity. Call this e0. You can add any number of elements to it. You can also take those things and for each create another arbitrary long number, w^e0. You can then go on and iterate this process. w^w^w...^e0. You can also look at e0^e0 and e0^e0^e0 and so on. You iterate this to e1. You can now do all this shit with e1 as well...
This is some ordinal but this is STILL COUNTABLE.
My point is, fixed size rows get you nowhwere on the infinity scale, no matter how you nest them Matrices are super tame.

https://en.wikipedia.org/wiki/Epsilon_numbers_(mathematics)

Eventually you get to ordinals that aren't countable anymore (in particular, the ordinal that holds ALL countable ordinals, w1), but as a matter of fact you run out of the possibility to describe higher and higher countable ordinals long before that
https://en.wikipedia.org/wiki/Church–Kleene_ordinal
That is to say, there's technically countable ordinals smaller that w1, but for them there's no notation possible.

Btw., if you have no issue with postulating the existene of e0, a countable collection of deep nesting of lows if you will (a sort of infinite tree), then you can get one element in correspondence with any Peano arithmetic proof and by induction proof Peano arithmetic consistent
https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof
(This doesn't fall for Gödels trap since it's not a theory of arithmetic in itself)