[spam][crazy][fiction][random] Non-Canon MCBoss Spinoffs
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Sat Sep 16 16:46:41 PDT 2023
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:12 PM
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<gmkarl at gmail.com>
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new unknown: diagonalization-lemma looks like obvious-formalization,
but understanding it further helps with issue with its use
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:13 PM
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great we can maybe study-repeat it
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:16 PM
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The diagonal lemma shows that in theories that can represent
computability, all formulas have a fixed point.
i don't think i agree wih that, maybe relates to formal definition of theory
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:17 PM
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what would it mean to not have a fixed point?
it w
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:18 PM
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so
selecting not working, but new unknown diag lemma: not obvious
formality. states every formula has fixed point.
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:20 PM
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wasn't using reference equality
diag lemma says g b g same g both sides
it asserts not that b and g can exist but that a g exists for every b.
strong claim
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:21 PM
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we expect diag lemma to cross logics
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:22 PM
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ybe mistake how absorb mistake
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:24 PM
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new unknown: "formula" in diag-def
1: The diagonalization of X is the formula (∃x)(x=⌈X ⌉ ∧ X )
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:28 PM
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the proof is broken into two lemmas
the first defines "diag(n)" (diag-def)
the second asserts formulas have fixed points
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:30 PM
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(it might be helpful to learn the formula syntax that uses an and?
symbol and looks a little like substitution
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:31 PM
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i figured it but itKs not stabilized in thinking
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:34 PM
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X is a formula containing x
diag(n) = ⌈(∃x)(x=⌈X⌉∧X)⌉
the expression means that -- waaaait !
I think it is saying that ... ohhhh
it's stating that x is X, and that the contents of X hold true, both
not sure why it says ∃x instead of something like ∃X but by the left
side they're equivalent
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(so lemma 1 is talking about fixed points)
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:38 PM
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lemma 1 says that if you can serialize unary formulas you can
serialize the expression of them having a fixed point in this way,
(and that they are "computable" new-unknown ctxual might just mean you
can solve it)
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:44 PM
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so: lemma 1 says fixed point expression (oops)
...the proof of lemma 2 parts looks obvious but don't see how
conclusion arises, have not connected all parts. some not understood,
looks relates "theorem copy-paste not work with whole at once
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Assume that D represents diag in T and let F be the formula (∃y)(D(x,y) ∧ B(y)).
i think this says that f says there exists a fixed point y in B
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thought: i'm guessing godel issolving his own expression here
general proof of fixed-pointness, if youhave a metamath youcan solve
the expression ofpassing theexpfession toitself, for the parameter.
requiresconcievingof where self-refis and whereparameter is andmoving
unknown to one side
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wait so um
x = f(
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:52 PM
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(we woulduse diff terms for describing domain andrange naybe confused
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i don't believeitithink metasummarywrongor misinterpreted
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 4:59 PM
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i'm havingtrouble understanding itbrelatedto retaning andcomparing the
summariesformed after considering theparts
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 5:11 PM
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> Assume that D represents diag in T and let F be the formula (∃y)(D(x,y) ∧ B(y)).
> Choose G ≡ (∃x)(x=⌈F⌉∧F) as the diagonalization of F and let n and g be the Go ̈del
new unknown: guessing that triple bars are formal definition, 1 check
> numbers for F and G, respectively. By definition of diag we know diag(⌈F⌉) = ⌈G⌉ and
> thus D(n_, g_ ) must be valid in T .
> Furthermore G ≡ (∃x)(x=n ∧ (∃y)(D(x,y) ∧ B(y)) is logically equivalent to the formula
> (∃y)(D(n_ ,y) ∧ B(y)). Because of the functionality of D and the validity of D(n_,g_) this
> formula is equivalent to D(n_, g_) ∧ B(g_), which in turn is equivalent to B(g_).
substituted g for y which was F which expands to similarity containing B?
> Thus G is logically equivalent to B(⌈G_⌉) in T and hence |=T G_ ⇔ B(⌈G_⌉).
> Note that the diagonal lemma holds for Peano Arithmetic, as diag is representable in any theory that can represent the computable functions.
it's notable godel and peers must have been holding famikiar intuition
to develop and argue this. used structures to describe/simplify
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Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 5:20 PM
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let's rephrase for clarity
> diag(⌈X⌉) = ⌈(∃x)(x=⌈X⌉∧X)⌉
...
> Let f and g be the Go ̈del numbers for F and G, respectively.
> Let F be the formula (∃y)(y=diag(x) ∧ B(y)).
issue
- Show quoted text -
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more study8ng methinks unless typing easier and can pin to one part
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 5:22 PM
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i don't know what D(x,y) means when diag has only one variable
it looks like it relates to the outside vs inside of theexprrssion
like one is n and the other diag(n)
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 5:26 PM
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> diag(⌈X⌉) = ⌈(∃x)(x=⌈X⌉∧X)⌉
> Assume that D represents diag in T and let F be the formula (∃y)(D(x,y) ∧ B(y)).
> Choose G ≡ (∃x)(x=⌈F⌉∧F) as the diagonalization of F and let f and g be the Go ̈del
> numbers for F and G, respectively. By definition of diag we know diag(⌈F⌉) = ⌈G⌉ and
> thus D(f_, g_ ) must be valid in T .
> Furthermore G ≡ (∃x)(x=f ∧ (∃y)(D(x,y) ∧ B(y)) is logically equivalent to the formula
> (∃y)(D(f_, y) ∧ B(y)). Because of the functionality of D and the validity of D(f_, g_) this
> formula is equivalent to D(f_, g_) ∧ B(g_), which in turn is equivalent to B(g_).
- Show quoted text -
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Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 5:31 PM
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> diag(⌈X⌉) = ⌈(∃x)(x=⌈X⌉∧X)⌉
> Assume that D represents diag in T and let F be the formula (∃y)(D(x,y) ∧ B(y)).
new unknown: D(x,y) ≡? diag(x) = y
- Show quoted text -
Add star mailbombbin<mailbombbin at gmail.com> Sat, Sep 16, 2023 at 5:38 PM
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- how did y turn to g_
- how is B(g_) isolated out of ∧
- Hide quoted text -
thinking about diag(n)
this stating that when a formula is substituted into itself, it is true
it is itself a formula that converts this statement to a godel number
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