Path: csiph.com!eternal-september.org!feeder.eternal-september.org!reader01.eternal-september.org!.POSTED!not-for-mail From: Ben Bacarisse Newsgroups: comp.theory Subject: Re: Simply defining =?iso-8859-1?Q?G=F6del?= Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability) Date: Thu, 30 Jul 2020 01:31:30 +0100 Organization: A noiseless patient Spider Lines: 157 Message-ID: <874kpp9731.fsf@bsb.me.uk> References:

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<87y2n3b8tz.fsf@bsb.me.uk> <87k0ynb3lq.fsf@bsb.me.uk> <46mdnVc5cLQdRL3CnZ2dnUU7-a3NnZ2d@giganews.com> <871rkubn07.fsf@bsb.me.uk> <87365a9lb6.fsf@bsb.me.uk> Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit Injection-Info: reader02.eternal-september.org; posting-host="877412a411aebc0b51c676483b2d6603"; logging-data="27508"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1993uJvUry3G13elIQtz7xbgZ6UKOE1fgg=" Cancel-Lock: sha1:hKENR03geonaQTxz3Hxj1Jt1frM= sha1:0lb4rTCg56+pb3/cFf5C9w0yPxo= X-BSB-Auth: 1.2e74946895d2cb5c66a8.20200730013130BST.874kpp9731.fsf@bsb.me.uk Xref: csiph.com comp.theory:22029 olcott writes: > On 7/29/2020 2:24 PM, Ben Bacarisse wrote: >> olcott writes: >> >>> On 7/29/2020 6:04 AM, Ben Bacarisse wrote: >>>> olcott writes: >>>> >>>>> On 7/28/2020 6:51 PM, Ben Bacarisse wrote: >>>>>> olcott writes: >>>> >>>>>>> except on the relation "father of" referring to an order pair >>>>>>> of unique individuals in the domain of humans. >>>>>> >>>>>> If you keep subtly altering what you say, my objection will be subtly >>>>>> different too. You will think I'm changing my mind (or being >>>>>> disingenuous) but I'm just responding to what you actually write. >>>>>> >>>>>> Here, I am glad to see you have written "father of" as if it were and >>>>>> ordinary English phrase. That's what Mendelson did, so I am happy to >>>>>> assume you mean the same relationship he did. >>>>>> >>>>>> If, instead, you make it look like a symbol (as you have done) and maybe >>>>>> give it arguments (especially arguments that look like constants in >>>>>> Mendelson's notation) it becomes a predicate or a formula and could mean >>>>>> anything depending on the interpretation. >>>>> >>>>> Not within the fricking context of our conversation. >>>> >>>> We are talking about the interpretation of formal languages. The >>>> conversation is about formulas, predicates, functions, constants, >>>> relations and domains. How could writing something the looks like a >>>> predicate (or maybe function, possibly a relation) when you did not mean >>>> to be anything other than confusing? I think your arrogance extends to >>>> thinking that anyone who does not know exactly what you mean must be >>>> dumb, disingenuous or a troll. >>>> >>>>>>> if {a26, a87} are a pair of baby girls then we can know that the >>>>>>> "father of" relation is not satisified by the ordered pair . >>>>>> >>>>>> You are free to make up your own notation, but you are using two symbols >>>>>> that Mendelson uses for constants. This can lead to confusion. Humans >>>>>> very often have names, so you could have said "if {Jane, Aisha} are a pair >>>>> >>>>> Yet the names are never unique. To specify unique individuals in the >>>>> universe we really need a GUID. (Globally Unique Idetifier). a26 and >>>>> a87 for a short-hand proxy for that. >>>> >>>> Then make them (look) unique without using confusing terminology: >>>> Jane82767, Aisha165. Remember, for a while, you actually claimed the >>>> constants are "in the domain of humans" so it was reasonable to think >>>> you were confused about what role constants play in all this. >>>> >>> >>> So more precisely: >>> The names of constants in L mathematically map to the the same names >>> of unique elements in D: L.a26 ↦ D.a26 and L.a87 ↦ D.a87 >> >> In other words, you know you were wrong, and you should have kept the >> distinction between the two clear. Obviously, being you, you have >> chosen to invent a new way of doing this that is deliberately open to >> misinterpretation. Here, you leave open the possibility of omitting the >> prefix: > > ∃!y ∈ humans ∃!x ∈ humans (Boolean father_of(x,y)) > > Where "father of" is to be understood to specify the same relation > between an ordered pair of humans that would be in English phrase: > "father of". Another notation! How many is that? Does !x replace your old (what, three hours old?) D.x notation? If so, what is the x without the !? Is it the old (three hours old) L.x? If so, father_of(x,y) is not a relation but a formula involving a predicate and we've left the interpretation again. And I love the poetic juxtaposition Boolean and father_of. Is it suppose to hint at a "return type"? Have you ever wondered if mathematicians already have a way of writing that? Did you think to ask? Does the specification of Boolean mean that this metaphorical maths of yours has relations that are not true/false? In fact, is it not a relation at all but simply a function? And is ∃!x ∈ S somehow more emphatically the case than the simpler and rather feeble !x ∈ S? Maybe the ! attaches to the ∃ in which case I've no idea how it works (it's not there later on in the line as I'd expect if it denotes an element of the domain). So many questions about just one short math poem. But rather than answer them, why not stop writing poems, pick up a book that does it property, and learn from it? >>>>> Except that some Janes are men with daughters. >>>> >>>> Don't come the raw prawn with me mate. You said they were "baby girls" >>>> as did I. This comment disappoints me. Up to know I tough you wer just >>>> confused. Maybe you are the one being disingenuous. >>> >>> I just wanted to make sure that we agreed on the concept of unique >>> elelments of a set. >> >> I want to think you are being dishonest, but maybe you are so ignorant >> of set theory that you think the elements of a set might not be unique. > > Yes I guess that they would be unique. I never needed to apply this before. > >>> The way that you said it would include some men named Jane that have >>> daughters. >> >> I wrote, copying your sentence, "if {Jane, Aisha} are a pair of baby >> girls then we can know that the "father of" relation is not satisfied by >> the ordered pair ". > > You did specify these details in your original reply so my critique of > this reply was incorrect. > >> There can be no other Jane in the >> domain other than that baby girl. There may be Jane2, Jane3 and Jane >> Doe, but we know for sure (if the antecedent is accepted) that Jane is >> not a man with children. >> So help me out. Was your reply prompted by ignorance of set theory, or >> was it a dishonest attempt to score a cheap point? > > My critique of your reply was incorrect. Also I was not aware that > every elements of a set is unique having never dealt with this aspect > of set theory before. Thank you. I will take your reply as honest despite the fact that you have had to deal with this aspect of set theory before. It came up when discussing your "triples" that were not triples but sets that might have fewer than thee members because set elements are always unique. >>> Other authors do it this way too: >>> https://sites.math.washington.edu/~aloveles/Math300Winter2011/m300Quantifiers.pdf >> >> That document says nothing about how to distinguish between constants in >> a formal language and elements in the domain of an interpretation. It's >> just some basic notation. > > It specifies how to tie variable of a formal language directly to > elements of a specific set: x ∈ ℝ In the formal languages we are talking about, there must be no set[1]. The whole point is to find out what sets might be models of a theory in that language. And when we are talking about a model, the set is understood. Writing anything but ∈ D (where D is the domain in question) would be an error. It just adds noise. [1] Actually that's not true but I hesitate to bring it up. The formal language could include restricted quantifiers (Mendelson does not do this) but then the ℝ or whatever just becomes a constant in the language whose interpretation in the domain of a model needs to be specified. -- Ben.