What are the achievements of Logic?

What is the basis for reason? And mathematics?

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wtf
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Re: What are the achievements of Logic?

Post by wtf » Thu Mar 08, 2018 9:11 pm

Dalek Prime wrote:
Thu Mar 08, 2018 12:56 am
I didn't see this post wtf, but again, I reiterate what I said about lamda calculus earlier. It did impact typed, but not untyped.
Perhaps you can elaborate a bit so that I can understand what you're trying to say. Lambda calculus has the exact same power as Turing machines so I'm not sure what exactly you are getting at. But if you are willing to provide some context without resorting to crude references to bodily functions, I'd certainly read with an open mind.

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Arising_uk
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Re: What are the achievements of Logic?

Post by Arising_uk » Thu Mar 08, 2018 11:16 pm

wtf wrote:...
The first incompleteness theorem simply says that given a consistent set of axioms sufficiently powerful to do mathematical induction, there are propositions that can neither be proved nor disproved. So no set of axioms can tell you everything that's true. There are always truths that lie beyond the power of any consistent set of axioms. ...
But I thought Godel proved that first order predicate logic was sound and complete? Is there a difference between mathematical induction and logical induction?

With respect to Godel's idea did not Russell already point this out with his paradoxes and supposedly solved this with his Type theory, i.e. self-referential propositions cannot be proved within an axiomatic system and a meta-logic must be used or are Gödel's propositions different from self-referring ones?

What I suppose I'm asking is why it's thought that Russell's work didn't ground Arithmetic in Logic due to Gödel's work?

wtf
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Re: What are the achievements of Logic?

Post by wtf » Thu Mar 08, 2018 11:39 pm

Arising_uk wrote:
Thu Mar 08, 2018 11:16 pm
But I thought Godel proved that first order predicate logic was sound and complete? Is there a difference between mathematical induction and logical induction?
Briefly, the competeness theorem says there's a model if and only if there's a proof. (Semantic truth = syntactic provability). The incompleteness theorem says that in any (suitably powerful) axiom system, there's a statement that can neither be proved nor disproved.

Here's a discussion of the subject. If I have more time later I'll try to explain this better but I can't do so at the moment.

https://math.stackexchange.com/question ... s-theorems
Arising_uk wrote:
Thu Mar 08, 2018 11:16 pm
With respect to Godel's idea did not Russell already point this out with his paradoxes and supposedly solved this with his Type theory, i.e. self-referential propositions cannot be proved within an axiomatic system and a meta-logic must be used or are Gödel's propositions different from self-referring ones?
No, Russell found the problem with unrestricted comprehension: defining a set using only a predicate. Rather, you either need type theory, or restricted comprehension. Restricted comprehension means that if you already have a set, you can cut it down to size with a predicate.

So the set of all sets that are not members of themselves leads to a contradiction. The set of all real numbers that are not members of themselves is just the real numbers. No contradiction. In the former case you have unrestricted comprehension; in the latter case, restricted.
Arising_uk wrote:
Thu Mar 08, 2018 11:16 pm
What I suppose I'm
asking is why it's thought that Russell's work didn't ground Arithmetic in Logic due to Gödel's work?
Not clear what you mean.

Dalek Prime
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Re: What are the achievements of Logic?

Post by Dalek Prime » Fri Mar 09, 2018 12:51 pm

Wtf, I was doing some reading last night, and have come to the realisation that I have been under misconceptions and ignorance of the subject matter. It was actually an article on the infinite which allowed me to see what you are talking about, and I thank you for challenging my notions. Having seen my errors in thinking, I want to apologize for being pig-headed about it, and hope you'll forgive any consternation I have caused. I like to think I'm still capable of learning and incorporating ideas new to me, and shedding misconceptions long held.

So, thank you. It's a constructive lesson and reminder of 'pride cometh before the fall'.

May I ask you then, if most finite systems escape Godel's incompleteness, and this effects only large systems trying to do too much ie. be 'everything'? I did note that Euclidean geometry seems clear of it. The cemtral issue is about provability, yes? Am i correct on this part?

alan1000
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Re: What are the achievements of Logic?

Post by alan1000 » Sun Mar 11, 2018 7:10 pm

This is supposed to be a philosophy site. In case you miss the subtlety of that, it means that if you disagree with someone, you are supposed to present contrary arguments. You may be surprised to learn that "Laughably incorrect" is not an argument. I'm listening...?

alan1000
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Re: What are the achievements of Logic?

Post by alan1000 » Sun Mar 11, 2018 7:16 pm

Sorry, Dalek Prime, my last was not a response to you - I forgot to include the quote. I was referring to an earlier post:

"But I saw your post on infinity over on that other site and virtually everything you say is laughably incorrect. You're just getting bad information and misunderstanding even that."

Frankly I'm beginning to question the value of these "philosophy" websites. There are too many posters who think that their prejudices, preconceptions and unsupported assertions carry just as much weight as a reasoned, objective argument.

wtf
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Re: What are the achievements of Logic?

Post by wtf » Sun Mar 11, 2018 7:28 pm

Dalek Prime wrote:
Fri Mar 09, 2018 12:51 pm
Wtf, I was doing some reading last night, and have come to the realisation that I have been under misconceptions and ignorance of the subject matter. It was actually an article on the infinite which allowed me to see what you are talking about, and I thank you for challenging my notions. Having seen my errors in thinking, I want to apologize for being pig-headed about it, and hope you'll forgive any consternation I have caused. I like to think I'm still capable of learning and incorporating ideas new to me, and shedding misconceptions long held.

So, thank you. It's a constructive lesson and reminder of 'pride cometh before the fall'.
Wow this never happens online Thanks for the post. Apology accepted.

Curious to know what you read that was so clear and compelling. I'd like to read it myself.
Dalek Prime wrote:
Fri Mar 09, 2018 12:51 pm
May I ask you then, if most finite systems escape Godel's incompleteness, and this effects only large systems trying to do too much ie. be 'everything'? I did note that Euclidean geometry seems clear of it. The cemtral issue is about provability, yes? Am i correct on this part?
Not sure exactly what you're asking. By finite systems do you mean finitely axiomatized? This might be outside my limited sphere of competence.

Euclidean geometry can't model the arithmetic of the natural numbers so it's not subject to incompleteness.

A more striking example is that the formal theory of the real numbers is decidable. Then you'd ask, but aren't the natural numbers a subset of the reals, so why aren't they decidable? The answer is that you can't actually pick out the natural numbers from the reals by a first-order statement. Please don't quote me on any of this but here's a nice Stackexchange question about it. See Pete Clark's checked answer.

https://math.stackexchange.com/question ... al-numbers

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