Have YOU verified the foundations of arithmetic?

>what is Euclid's Elements

Yes, and I found them lacking.

The issue I have with them was introduced in the late 1800's/early 1900's. When logic was being formalized. Personally I'm surprised it wasn't caught considering many related results at the time.

Foundations are related to proof and theorem. You are correct that mathematics has been around longer than we've had axiomatization but that doesn't mean that modern mathematics is practiced without proof or theorem.

Shit, by modern standards.

Fucking wew that identity property, i sincerely doubted it would work.

the denominator of the zero is distributed to both sides

Yeah. From Peano.
Got bored part way through some order axioms

Im curious as to your thoughts after that determination (that math is gay)

Fyi im following wildberger which ive surmised is fringe. He has a lot to say about established set theory etc

have YOU seen us.metamath.org/index.html ?

Extreme gratitude to you user!

I hope you're actually using induction in your proofs. Many of the proofs for natural and rational arithmetic requires multiple nested inductions and they can be fairly nontrivial.

I used a combination of axiomatic set theory books and Landau's foundations of analysis. The problem I encountered affects every approach to the reals within set theory.

I don't think math is gay. I just think it's important to realize the limitations of formal languages when dealing with uncountable sets.

Mild appreciation of expression of extreme gratitude towards self.

i would but i'm sick of studying this shit. gonna get my easy ms and a comfy job.

i think im using induction (i have a nonsci undergrad and im just seriously getting into mathematics). I believe the only assumtion ive made is that natural numbers are countable/orderable entities, and now ive gotten to a rational number field.

>uncountable sets
im a few vids away from wildbergers assertion that the infinity set doesnt exist, which im anticipating with much excitation. Its math foundations ~16 or so on his youtube channel, njwildberger, if youd like to see what im seeing/following along with. He prefaced the series with the idea that he has a better approach, vids are now near #200 many of them 30+ minutes. I hope to catch up by 2017

I remember doing that in first semester of university after introducing peano axioms
most of the stuff you are doing in your picture seems pretty redundant though

It's self-evident.

I verified them experimentally in about 3rd grade.

what is \ ????

I was using \ for 'less'. That first pic is for integers where an integer is two natural numbers a\b. Its wildberger's approach to negative numbers.

He defines rationals as a set of integers a/b. Zero as a rational number ends up being (1\1)/z where z is a non zero integer.

While many aspects are self evident, there are some very interesting things i have gleaned from this exercise. The most interesting has been the pattern/relationship between the basic operations

utterly wasting your time

>have YOU verified the foundations of arithmetic?

Don't need to. It's part of the standard library: github.com/coq/coq/tree/trunk/theories/Arith

>not writing your own libs

Im learning a lot breh. Later ill CAD a deeper visualization of ford circles to see the more complex/long manifesting patterns, just for laughs.

Also i hope you like multicoloured notes and norman wildberger, because i plan to post this shit as i study it. For discourse, though.

bump

>these have no relevance to actual mathematics

For most practical applications I'd agree with you. But for analysis and the justification of otherwise infinitesimal calculus and such it's nice to make sure that what you're relying on works out.

>verified the foundations of arithmetic
we have proven already that this is impossible. see goedels incompleteness theorem

Whats the total area inside all of those ford circles then? You can probably give the answer, but most definitely not verify it since u seem to be wasting your times verifying the axioms of arithmetic. Be a real nigga and fuck with modular forms n shit instead of doing this pointless shit