The Road to Gödel's Incompleteness Theorems - Juliette Kennedy

Friends Lunch with a Member

Topic: The Road to Gödel's Incompleteness Theorems
Speaker: Juliette Kennedy
Date: November 22, 2019

[BlueSoulTiger]

Any chance that the question+answer session can be published please IAS? Such sessions are often more insightful than the main talk.

[SuperCuteAnimeWaifu]

I like how like 10 years after the guy who created that quote "god created the integers...." every good mathematician shat on it once. I think non rigorous mathematical lectures is a nice ambient noise to eat to. Thank you.

[rg]

The fact I can think anything at all is absolute proof that there cannot be an infinity of anything. For if there were infinity of anything, its informational content would be so vast that it would make my mind impossible to exist apart from that infinity. An infinity of anything would leave absolutely zero space in the universe for something else to exist such as my own finite mind.

Anyone can say there is an infinite number of this or that, and attempts were made to escape the impossibility by using shortcuts like one to one correspondence, isomorphisms, but there is always a tacit presumption that there exists an infinite storage space that 'could if it exists' store such large numbers.

I think counting, I think the natural numbers, is not a reference to what could exist outside of us, but is a product of the frequency or iteration pattern generation of self-referencing strange loop consciousnesses like our own. When we count 1, 2, 3, ..., I think what is happening is that there is a finite set of loops of perhaps the most basic action of all, positing the self and then doing it again while storing the previous self posit in memory. When we count the number of apples in the basket, we are simultaneously counting our self-posit iterations and 'mapping' those self-posits to each physical object.

I think the proof that there are an infinite number of prime numbers is not really a proof that there are an 'infinite' number of prime numbers, only that the human mind, and likely any other mind, cannot list them all in any finite set, which our brains THEN process as 'meaning the same thing' as infinity. When we posit ourselves 1, 2, 3, etc times, and when we map each iteration to a set of objects outside ourselves, it's easy to 'lose ourselves' and imagine natural numbers going on for infinity 'somewhere out there'. I think the natural numbers aren't 'natural' at all, they are symbols we assign to a set of memory stored by a self-referencing consciousness positing itself in a singular unit frequency. I exist, I said it once, I say it again, and again, and before I know it, I'm able to count apples, or Godel numbers.

I can prove there is not an infinite number of prime numbers using similar logic that Euclid used to prove there must be an infinite number: Suppose there are an infinite number of prime numbers. Then, I should be able to posit a prime number that is so large, that the informational content of the number equals the informational content of the entire universe. But the fact there exists information in the universe NOT a prime number, namely this youtube comment right here, the existence of this comment is 'proof by contradiction' that arbitrarily large prime numbers cannot in fact exist.

Or, another way of 'proving' this: Suppose we assign a 'RG number' code to the entire universe. It's like a Godel number, but named after me. This number let's suppose has exactly the same number of 'bits' as does the entire universe. Even though I just finished saying even that is an impossibility, I am offering it as a supposition. Now even if the entire universe could be coded as one gigantic number, it is impossible for there to be any greater number, because by definition the universe includes all information encoded by that RG number. There cannot be any primes larger than this number, for such numbers would contain more information than exists in the entire universe. Therefore there is a maximum prime number. Even if were to assign 'RG' to it, and use Euclid's reasoning, we could not even posit RG + 1, or RG plus anything, without instant contradiction because RG is by definition the numerical code for all of the information in the entire universe. If we supposed RG + 1 did exist, it would have to be in the enumerated set/number RG, which we defined as the code for all information in the universe. But that would be a contradiction, so our attempt to create 'RG+1' is an error.

The two thousand year old notion that because our human minds can count natural numbers but cannot comprehend a 'final' end to such counting, that this supposedly means there must be an infinite number of natural numbers, is I submit based on an error in assuming there is an infinitely large 'hard drive' that can save an infinite number of numbers in the universe.

There cannot possibly be an infinite number of anything, because of there was, that which is claimed as infinite would informationally take up the entire universe.

There is in fact a finite number of everything. What we call or think of as infinity, is our minds, and every other mind in the universe, coming to grips with the fact that there are some numbers so incredibly huge that they could not 'fit' into the universe let alone in a subject, mind or hard drive.

For Godel's incompleteness theorems, Godel never applied HIS OWN logic to his own theorems. He applied mathematical reasoning to mathematics and showed that no finite rich enough formal system can be both complete and consistent. OK, so much for systems like PM and ZFC. But what about Godel's own proof? By his own logic, his proof cannot be both a complete description of the natural numbers and internally consistent. Either Godel's theorems are missing something about the natural numbers, or, his theorems are inconsistent. I submit that Godel's theorems, if they are considered a 'complete' truth about all sufficiently rich enough formal systems, they must be inconsistent. Or, if they are consistent, they cannot be complete.

If you imagine the entire universe as consisting entirely of Planck cubes of information, where each Planck cube has a finite amount of information, we can imagine selecting an RG number code so incredibly big that it would uniquely 'code' all Planck cubes together in the universe. It would be impossible for there to be any more information available in the universe to construct a number larger than this RG number. It would be like asking a 1 terabyte hard drive to store and reproduce a number that requires more than 1 terabytes of information.

You could try to write down RG+1 again, using crude 'hey by using axiom X, I can add one to any natural number to get a larger number', but that would be an error because if RG+1 can be expressed verbally, it is by definition already in the set of information encoded by RG, so RG+1 can't exist apart from RG. Uttering ANY string is by definition already included in the RG number code for the entire universe.

The largest prime is so gigantic that it must necessarily escape every finite algorithm that posits prime numbers. There will always appear to be an infinite number of primes if the false assumption is made that there is an infinite informational hard drive to store it, which is what every proof since Euclid have assumed.

Using a mathematics that does not exist yet, I think one day we may see mathematical conclusions that look like "There exists a maximum prime number that cannot be computed by any algorithm equal to or smaller than the entire universe", and "For any sufficiently complex formal 'meta-system', like Godel's, that proves a formal system, like PM, cannot be both complete and consistent, that meta-system must itself be either complete and inconsistent, or consistent and incomplete, but not both complete and consistent."

[benheideveld]

A final example: the famous Theorem: “This Theorem is false”. If it is false it is true and if it is true it is false, right? Because every theorem is either true or false, right? Wrong! I’m a constructivist intuitionist, a strong Dutch Brouwerian tradition, and I live in Amsterdam just like he did. The crux is that the sentence is NOT A THEOREM. Therefore the paradox evaporates in front of our eyes. The proof that it is not a Theorem goes as follows. The phrase “This Theorem” (shortened as TT) is a placeholder that represents the whole theorem. So we must substitute TT for “TT is false” yielding: “TT is false is false”. But also this sentence still contains TT. Substituting again yields:
“TT is false is false is false” or with brackets:
(((TT) is false) is false) is false).
After unlimited substitution it is revealed that nothing is claimed by this construct. Therefore it is not a Theorem. Therefore it needn’t be either True or False. QED.

[naimulhaq]

Godel never proved anything mathematical. Logic (infinite axiom) and mathematical logic (finite logic) are different. The problem is with 'infinity', still undefined just like Russell's set of all sets. Divine design contains infinity within the finite.