It avoids the infinity symbol - it doesn't avoid infinity. It avoids the denotation of infinity, not the connotation of infinity.
I wonder: How long does it take the average Mathematician to perform an infinite number of summations?
Well, you have to accuse me of "lacking understanding" least you show yourself to be a fool who has fallen into the trap of circular reasoning.
The above is only true if and only if the Sigma function is total. It would be true a posteriori the function having converged. It is not true a priori while the function is busy converging.
I am pointing out to you that functions with infinite inputs are not provably total, because functions with infinite inputs are called Turing machines.
Obviously, you can choose to pave over the halting problem and declare them total so you can practice your religion, but at that point you and I have nothing to say to each other. You have an Oracle machine. I don't.
I mean provably total! I mean "your Sigma-function will actually halt". And by 'halt' I mean the exact same thing Church and Turing meant.
I mean your function terminates and produces an answer, as opposed to "runs forever" if you were to actually evaluate it using pen and paper.
1. Curry-Howard isomorphism. Syntactically e.g denotationally speaking proofs are programs. To prove your series - write me an algorithm which evaluates it to 1.
2. Provable totality requires Walther recursion which mandates finite inputs.
Your can't implement your sigma function using Walther recursion because you have an open interval. [1, ∞)
3. You can't implement this using an iterator either, because iterating over infinite sets is NOT a total function!
This is what it looks like in practice https://repl.it/repls/VitalCompatibleOffice
Code: Select all
def sigma(n):
if n == 1:
return float(9/10**n) # HALT!
else:
return float(9/10**n) + sigma(n-1)
print(sigma(n=1)) # 0.9
print(sigma(n=2)) # 0.99
print(sigma(n=3)) # 0.99
print(sigma(n=16)) # 0.9999999999999999
It is obvious then than sigma(n) = 0.99999999999999999...
It is also obvious that sigma(∞) DOES NOT HALT
Well, I am not a Mathematician, so I am not trying to get Math off the ground - that's your problem. I am an engineer, so I am very much trying to keep Mathematics grounded. You aren't my high-school teacher - I am not here to convince you of anything.wtf wrote: ↑Tue Oct 22, 2019 12:55 am Every form of constructive or neo-intuitionistic math accepts at least some form of AC, since without it you can't get much modern math off the ground. Even quantum physics depends on functional analysis which depends on a weak form of AC. I wonder if you know any of this. You haven't convinced me.
I accept the abstract notion of choice, and the abstract notion of a function. In particular, I accept no Mathematical function as being a magical black box. If your choice-function is one of those - it goes in the trash can.
I am happy to accept any choice-function you can realise/prove using the exact same type-theoretic notion of 'proof' introduced by Curry-Howard. If you can write an algorithm for your choice-function in Lambda calculus - you have proven your choice-function. This goes for your cardinality/ordinality functions too. Don't assume them - prove them by realising them.
I am really keen to see what kind of total function you are going to write for randomly choosing elements from infinite sets. I have so many questions , but the most nagging one being: "What is the sample mean (with and without replacement) of the Integers?"
I am super-keen to see your theoretically proven random number generator!
No. I think I am in a world which obeys Bremmerman's limit.
Mathematics is symbolic - like all religions.
Because if the Naturals are countably-infinite, you are going to run out of universe before your series actually converges to 1.
All "completed" infinities are nonsense. A completed infinity immediately becomes a symbolic infinity, not an actual infinity.
When it comes to ∞ syntax is not semantics. The fucking etymology of the word "definition" comes from root "finite".
