Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 4:12 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:13 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> Reply | Reply to all | Forward | Print | Delete | Show original great we can maybe study-repeat it Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 4:16 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:17 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> Reply | Reply to all | Forward | Print | Delete | Show original what would it mean to not have a fixed point? it w Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 4:18 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:20 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:21 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> Reply | Reply to all | Forward | Print | Delete | Show original we expect diag lemma to cross logics Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 4:22 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> Reply | Reply to all | Forward | Print | Delete | Show original ybe mistake how absorb mistake Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 4:24 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:28 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:30 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:31 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> Reply | Reply to all | Forward | Print | Delete | Show original i figured it but itKs not stabilized in thinking Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 4:34 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:35 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> Reply | Reply to all | Forward | Print | Delete | Show original (so lemma 1 is talking about fixed points) Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 4:38 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:44 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:48 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:50 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 4:51 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> Reply | Reply to all | Forward | Print | Delete | Show original wait so um x = f( Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 4:52 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> Reply | Reply to all | Forward | Print | Delete | Show original (we woulduse diff terms for describing domain andrange naybe confused Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 4:52 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> Reply | Reply to all | Forward | Print | Delete | Show original i don't believeitithink metasummarywrongor misinterpreted Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 4:59 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> Reply | Reply to all | Forward | Print | Delete | Show original i'm havingtrouble understanding itbrelatedto retaning andcomparing the summariesformed after considering theparts Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 5:11 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 5:20 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 5:20 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> Reply | Reply to all | Forward | Print | Delete | Show original more study8ng methinks unless typing easier and can pin to one part Add star mailbombbin<[email protected]> Sat, Sep 16, 2023 at 5:22 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 5:26 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 5:31 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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<[email protected]> Sat, Sep 16, 2023 at 5:38 PM To: "Undescribed Horrific Abuse, One Victim & Survivor of Many" <[email protected]> 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
