PDF with the definition of Q
Here is the definition of Q in Letter Notation:
Definition of Q - PDF
I'm only posting this since I promised (and I will not be participating in the discussion here).
Question to the experts: Is this statement really false?
Recently P-bot told me that the statement:
"Godel's [2nd] incompleteness theorem implies both of NBG+Con(NBG) and NBG+Neg(Con(NBG)) are consistent as long as NBG is consistent."
Is false.
And I don't understand why. Doesn't the 2nd incompleteness theorem state that Con(X) is undecidable in X, assuming X is consistent? And doesn't undecidablity immediately implies the bolded statement above?
What am I doing wrong here?
EDIT 1: Due to abusive replies, comments have been disabled on this blog post.
EDIT 2: Got it! It was a simple oversight on my part which can be explained in a single line: "The fact that statement Y is unprovable in X only implies X+Neg(Y) is consistent.
An even shorter Rayo name for 65536: Who needs ordered pairs?
In my previous attempt to create a record Rayo name for 65536, I've used classic definition of ordered pairs ((x,y)=.
Ordered pairs are quite symbol-extensive, so it would be nice if we could find a way to circumnavigate their use. This can indeed be done, by replacing (x,y) with the ordinary doubleton {x,y} and using other means to ensure that this doesn't lead to ambiguity.
The idea is simple:
1. When seeking a bijection from set A to set B, we will notate the ordered pairs as {x,y} where x∈A and y∈B.
2. If x∈A and x∉B then the meaning of {x,y} is non-ambigious: x must be the representative of A and y must be the representative of B.
3. If x∈A⋂B the we will always pair it with itself. In other words, we will only be interested in bijections t…
A Rayo name for 65536 under 850 symbols (!)
To everyone: I would really appreciate if you check my work and point out any errors.
This is based on the previous work both by P進大好きbot and myself, as well as a completely new (and much more efficient) Rayo string that represents bijections.
Also special thanks to Emk who corrected the expression for Transitive sets (r10 and r11) while keeping the total length unchanged.
EDIT:
We can do a bit better, by using a variation of an idea by Emk and defining a certain kind of 4-element set, to replace the number 2:
∃b∃c∃d∃e(b∈y∧c∈b∧d∈c∧e∈d∧c∈y∧d∈y∧e∈y∧¬∃f(f≠b∧f≠c∧f≠d∧f≠e∧f∈y))
This definition costs 104 symbols, which is 26 less than the definition of y=𝒫(2).
So we get:
Rayo(784)>65536
Note: Due to the amount of typos and miscounts that were already found…
Proof that Rayo(n) is 1 for n between 10 to 19
The basic insight on which this proof is based, is that we can safely ignore any substrings which are either tautologies or contradictions. It's easy to show that such substrings either has no effect on the compound string, or they turn the compound string itself into a tautology/contradiction:
Let A,B be tautologies.
Let C,D be contradictions.
Let E be any other expression.
Then using Rayo's rules, we can build the following 9 statements (and only them):
(a) (¬A) which is a contradiction.
(b) (¬C) which is a tautology.
(c) (A∧B) which is a tautology.
(d) (A∧C) / (C∧A) which is a contradiction.
(e) (C∧D) which is a contradiction.
(f) (A∧E) / (E∧A) which is equivalent to E.
(g) (C∧E) / (E∧C) which is a contradiction.
(h) ∃xi(A) which is a tautology (if…