17 Sep
2023
17 Sep
'23
8:46 a.m.
Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:12 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original new unknown: diagonalization-lemma looks like obvious-formalization, but understanding it further helps with issue with its use Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:13 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original great we can maybe study-repeat it Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:16 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original 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@gmail.com> Sat, Sep 16, 2023 at 4:17 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original what would it mean to not have a fixed point? it w Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:18 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original so selecting not working, but new unknown diag lemma: not obvious formality. states every formula has fixed point. Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:20 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original 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@gmail.com> Sat, Sep 16, 2023 at 4:21 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original we expect diag lemma to cross logics Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:22 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original ybe mistake how absorb mistake Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:24 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original new unknown: "formula" in diag-def 1: The diagonalization of X is the formula (∃x)(x=⌈X ⌉ ∧ X ) Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:28 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original 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@gmail.com> Sat, Sep 16, 2023 at 4:30 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original (it might be helpful to learn the formula syntax that uses an and? symbol and looks a little like substitution Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:31 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original i figured it but itKs not stabilized in thinking Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:34 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original 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 Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:35 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original (so lemma 1 is talking about fixed points) Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:38 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original 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@gmail.com> Sat, Sep 16, 2023 at 4:44 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original 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 Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:48 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original 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 Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:50 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original 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 Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:51 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original wait so um x = f( Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:52 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original (we woulduse diff terms for describing domain andrange naybe confused Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:52 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original i don't believeitithink metasummarywrongor misinterpreted Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 4:59 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original i'm havingtrouble understanding itbrelatedto retaning andcomparing the summariesformed after considering theparts Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 5:11 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original > 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 Reply | Reply to all | Forward | Print | Delete | Show original Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 5:20 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original 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 - Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 5:20 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original more study8ng methinks unless typing easier and can pin to one part Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 5:22 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original 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@gmail.com> Sat, Sep 16, 2023 at 5:26 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original > 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 - Reply | Reply to all | Forward | Print | Delete | Show original Add star mailbombbin<mailbombbin@gmail.com> Sat, Sep 16, 2023 at 5:31 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original > 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@gmail.com> Sat, Sep 16, 2023 at 5:38 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <gmkarl@gmail.com> Reply | Reply to all | Forward | Print | Delete | Show original - 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