Mathematicians are in deep shit for 2 reasons

[Ed3]

Hi Ladyjane,

“I am shocked! Shocked! I say. To find …” someone attacking a rival’s proof.

Godel’s theorem presently stands even if Godel’s proof is invalid, BECAUSE Peter Smith’s proof, which is totally different than Godel’s proof, is considered valid.

At this point I am torn as to whether or not I should rehash my comments on the Skolem Paradox, but I will try again assuming that there may be other people reading this post.

Skolem’s Paradox says that if axiomized ZFC has a model, then (because ZFC is a first order theory) by the Lowenstein Skolem theorem that model is countable. Since ZFC implies uncountable sets exist, the model must be uncountable.

In order for this paradox to be valid there must be a model for ZFC. More particularily there must be a model for the the axioms of ZFC.

This means that there must be a set from which we can construct , using the approved grammer, the axioms of ZFC; and most importantly (by the definition of model) we are required to prove that these axioms are true.

Even though a couple of axioms of ZFC are in fact theorems, in general they are not. For poetic justice I will asssign you the task of proving that the axiom schema of comprehension is true .

All of the talk about isomorphisms, countable, and uncountable sets rests on finding this model.

Never say never, but it ain’t gonna happen.

[ladyjane]

In order for this paradox to be valid there must be a model for ZFC. More particularily there must be a model for the the axioms of ZFC.

i say LST says there is a model

from wiki

NOTE IT SAYS WE CAN GET A MODEL
quote

en.wikipedia.org/wiki/ZFC

Zermelo–Fraenkel set theory, with the axiom of choice, commonly
abbreviated ZFC, is the standard form of axiomatic SET THEORY

NOW read wiki

Using the Löwenheim-Skolem Theorem, WE CAN GET a model of SET THEORY
which only contains a countable number of objects. However, it must
contain the aforementioned uncountable sets, which appears to be a
CONTRADICTION

now ZFC is set theory
so it is in contradiction due to skolem- thus inconsistent

Godel’s theorem presently stands even if Godel’s proof is invalid, BECAUSE Peter Smith’s proof, which is totally different than Godel’s proof, is considered valid.

this has nothing to do with smiths proof
even wiki says godels theorem is about truth ie true statements and godel cant tell us what true statements are thus his theorem is meaningless

en.wikipedia.org/wiki/G%C3%B6del … s_theorems

Gödel’s first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. 

[Ed3]

Hi Ladyjane,

Much to my surprise, your references concerning models of ZFC are interesting, and do in fact contradict my assertion. I need to read more on the subject to be convinced, but your references are impressive.

However, I still think that you don’t understand the implications of Smith’s proof.

[ladyjane]

hi Ed3

you say

However, I still think that you don’t understand the implications of Smith’s proof.

it the implications of godel not telling us what makes statements true which is the issue
and colin leslie dean points out it mean his theorem is meaningless

you say

However, I too disapprove of ZF axiomatic set theory.

The reason for this is that there are at least two axioms which are not axioms at all. They are theorems. Additionally, one of those axioms has the misnomer of “Axiom of Infinity”. This theorem reads that there exists an inductive set. What were they thinking???

I have other personal problems with ZF but I am in the minority in those regards.

i dont doubt that what you say is true

i might be wrong but i think skolem paradox arise because of the axiom of infinity
if as you say it is not an axiom
then skolems paradox vanishes- but then does ZFC axiom system to

[IrvingWashington]

Not that I know anything about mathematical logic, but a general point. Do people really argue by citing wikipedia these days? That is ludicrous.

Ladyjane, you still haven’t answered the very simple point that ‘mathematicians’ are people who do maths, and problems in the foundation of maths and philosophy behind it mean little to a working mathematician. I study applied maths and applied mathematicians are happy enough talking about ‘dividing’ by derivatives, so I don’t think they could care less about our little philosophical troubles. As such, mathematicians are in no kind of deep shit.

For mathematicians to be in deep shit we’d have to start adding 3 apples and 4 apples and getting 8 apples.

I’ve heard it said that the genius of Hume was that he managed to prove conclusively that its very hard to prove anything conclusively. I would generalise this to being the genius of philosophy in general. The foundations of maths being no exception.

[ladyjane]

irving says

Ladyjane, you still haven’t answered the very simple point that ‘mathematicians’ are people who do maths, and problems in the foundation of maths and philosophy behind it mean little to a working mathematician. I study applied maths and applied mathematicians are happy enough talking about ‘dividing’ by derivatives, so I don’t think they could care less about our little philosophical troubles. As such, mathematicians are in no kind of deep shit.

i say
this is a philosophy forum
if you cant see what colin leslie dean is cliaming is interesting philosophically then why be on a philosophy forum - and what he is pointing out is that mathematics is meaningless even though it works

as dean has said mathematician are a dumb lot
they can add the 3 + 4 apples and get 7 apples
but they dont know what any of it means

[IrvingWashington]

But the point of mathematics, well applied mathematics, is that you use it to model the world. We can model how the sun works, for example. This allows us to understand why the sun behaves as it does. The difference with physics is that you completely understand that all you are working with is a model. But, anyway, the point is that mathematicians can tell us exactly what an equation describing the magnetic field of the sun means. It means that our model of the sun operates in a particular way. If our model is any good this tells us something about the world. This is good.

You’re going around making ridiculous comments like ‘mathematics is meaningless’. I’ve just told you how mathematical equations can have meaning. There are different types of meaning, Ladyjane. I am not just making a pragmatic point, though I’d make the argument that maths works too. Mathematics is a broad term. While you may think you can reduce some parts of maths to logic, surely nobody thinks that what applied mathematicians do reduces to logic. If a piece of applied mathematics describes the world well, then it is meaningful, no matter what the foundations are.

So mathematics isn’t all about logic. That was just a conceit of particular philosphers. That is why mathematicians aren’t in deep shit, this is a philosophical forum, and this is a philosophical point.

[ladyjane]

You’re going around making ridiculous comments like ‘mathematics is meaningless’. I’ve just told you how mathematical equations can have meaning.

useing those equations the scientific model of the world ends in meaninglessness ie contradiction

Heisenberg notes that “ the strangest experience of those years was that the paradoxes of quantum theory did not disappear during this process of clarification; on the contrary they have become even more marked and exciting.”

In regard to the paradoxes and contradictions of quantum theory Wick state the orthodox view when he says “here my opinion of the orthodox quantum mechanics, like Bohr, comes down to the meaning of words. “Classical” and “complementarity”, insult and commendation, are euphemisms; the belief concealed is that Nature has been found in a contradiction. But quantum physicists are not simpletons. In their hearts they know such a claim is philosophically unacceptable and would be rejected in other sciences.”
Wick notes “ I believe orthodox quantum theorists [slates] reason, consciously or unconsciously, something like this. The microscopic world exhibits paradoxes or contradictions and this fact is reflected in the best theory describing it.”

when mathematician investigate the foundations of math ie why how it fits together

their explanations end in meaningless as dean has shown

yes mathematician can model the sun
but
why any of it works is a mystery as the explantion given end in meaninglessness

we are just playing with our selves in the dark

[IrvingWashington]

Look, if you’re going to just restate your opinions without engaging with what I’ve written there’s no point in this. I just stated that there are types of meaning. Lets accept your conclusions: mathematics ends in logical meaninglessness. That doesn’t mean it doesn’t have physical significance. If an equation describes how something in the world operates correctly then it has meaning just in virtue of doing that. Stop reducing everything to logic.

How does the Wick quote help you? It states that maybe the world is logically contradictory. Then why is it an objection to anything that it ends in logical contradiction? If, as it aims to, maths models the world accurately then we should expect logical contradictions. Maths is in order.

Why should there be an explanation for why our model of the sun works? You’re asking one question too many. I mean, we could have an explanation of the reason for each part of our model, we could talk about physical significance, what kind of answer would you want for the question ‘why does our model work’?

[Ed3]

Hi Ladyjane,

The short answer is that I don’t think that the Axiom of Infinity has anything to do with the Skolem Paradox.

Here I will elaborate on why the Axiom of Infinity is a theorem. The Natural numbers N are defined by a sequence of sets as follows:

Φ is 0
{Φ} is 1
{Φ, {Φ}} is 2
{Φ,{Φ},{Φ,{Φ}}} is 3
and so forth, where Φ is the empty set.

Here each new number is obtained by creating a new set, which contains all the elements of the preceding set except that it has an additional element that consists of the set of all elements preceding it.

There are only two axioms required to produce the Natural numbers. One is the axiom of existence that says that Φ exists, and the axiom schema of comprehension that allows the other sets to exist.

The Axiom of Infinity states: “An inductive set exists”

A set S is defined to be inductive if:
0 is an element of S and if n is an element of S then (n + 1) is an element of S.

0 is an element of N and if n is an element of N then n + 1 is in N.

Therefore the axiom of existence and the axiom schema of comprehension imply that an inductive set exists the (Axiom of Infinity).

There are inductive sets that are finite. Some examples are the sets denoted as N mod A. N mod 2 consists of the set {0,1}. Here 0 + 1 is defined as 1 and 1 + 1 is defined as 0. This is analogous to Odds and Evens. N mod 3 is the set {0,1,2} with addition (technically, the successor function) defined by 0 + 1 = 1, 1+ 1 = 2 and 2 + 1 = 0. It is easy to verify that these sets are inductive.

So not only is the Axiom of Infinity not an axiom, but it does not even require infinite sets to exist.

A final thought related to a different post of yours:
You should consider using the logical statement -(PV-P) in your signature.

[Ed3]

Hi IrvingWashington,

I have agreed with your comments about the wave particle duality and your observation that mathematics has little to do with the Godel Theorems.

It’s nice to someone with some math background on the site.

Thanks Ed

[ladyjane]

Ed3 said
Much to my surprise, your references concerning models of ZFC are interesting, and do in fact contradict my assertion. I need to read more on the subject to be convinced,

so are you convinced ZFC is inconsistent

[Ed3]

Hi Ladyjane,

I do not believe that ZFC is inconsistent. However, I do think that it is immature, and I am now interested in the Von Neumann–Bernays–Gödel set theory (NBG) referenced in your link:

en.wikipedia.org/wiki/ZFC

If you recall my post “Musings on the Skolem paradox”, you might remember that I said that the only way that I could see of finding a model for ZF was to find an alternative axiomization for set theory. This might do the job.

This link is very thought provoking and I very much appreciate it.

Thanks Ed

[IrvingWashington]

@Ed3

Hi IrvingWashington,

I have agreed with your comments about the wave particle duality and your observation that mathematics has little to do with the Godel Theorems.

It’s nice to someone with some math background on the site.

Thanks Ed

Thanks, I don’t know anything about mathematical logic though. I stay away from pure maths entirely. Applied mathematicians have all the fun.

[MagsJ (Surya Loka)]

@IrvingWashington

Thanks, I don’t know anything about mathematical logic though. I stay away from pure maths entirely. Applied mathematicians have all the fun.

I was studying ‘Pure and Applied maths’ at A-level grade: I guess it encompassed the best of both worlds - can you not enrol on such a course in the USA/in your state?

[IrvingWashington]

I’m in Britain. I’m studying maths and philosophy at university, at least until next month. I’ve just chosen to stay away from pure maths because I don’t like it. I’d far rather learn how the sun works, which is what I’m doing right now.

[MagsJ (Surya Loka)]

@IrvingWashington

I’m in Britain. I’m studying maths and philosophy at university, at least until next month. I’ve just chosen to stay away from pure maths because I don’t like it. I’d far rather learn how the sun works, which is what I’m doing right now.

The ‘pure’ part was pretty boring: the fun part kicked in in the application of it, so you aren’t wrong there!

[denali]

@ladyjane

Godel doesn’t need to tell us a damn thing about what’s true,

sorry if he cant tell us what true is then we dont know what any of this means
and it then is meaningless babble

if i said there are gibbly statements which cant be proven with out telling what makes a gibbly statement
you would say i am talking meaningless nonsence

thus with out telling us what makes a statement true all this is meaningless nonsence
en.wikipedia.org/wiki/G%C3%B6del … s_theorems

Gödel’s first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves basic [b]arithmetical truths,[/b] an arithmetical [b]statement that is true,[/b] but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. 

Godel did is find a general method that enabled him to take any theory T
strong enough to capture a modest amount of basic arithmetic and
construct
a corresponding arithmetical sentence GT which encodes the claim ‘The
sentenceGT itself is unprovable in theory T’. So G T is true if and only
if T can’t prove it

If we can locate GT

, a Godel sentence for our favourite nicely ax-
iomatized theory of arithmetic T, and can argue that G T is
true-but-unprovable,

See ladyjane, here’s where I win - you just quoted wikipedia interpreting a result. IIRC, the actual conclusion is a bit more like: Let A denote the set of all “provable” statements in a system P which models our notion of arithmetic; then there exists a statement G such that G is not in A if P is consistent.

Whether or not G is “true” is clearly not determinable or meaningful in the system by the very nature of the proof. If you want to take a formalistic approach to mathematics, all that Godel showed is that either P is incomplete (in that there are well formed statements which cannot be proven) or P is inconsistent. In other words: you’re attacking a popular position related to the proof rather than the proof; if you could just clean up your language on this point I would be more than willing to stop pointing out your error.

On top of that, saying that the argument “G is true though G is not in A” is “meaningless” is invalid; all we have to do is generalize our notion of “truth”. Or do any number of other things. Burden of proof is on you to show that we have no alternatives but to declare it meaningless, and to do so you’re going to need to assume a particular definition of “truth”, in which case we’re playing language games anyway as far as I’m concerned. You can’t invalidate mathematics without assuming an equal quantity of BS on your own part.

[ladyjane]

Smiths account is not popular and he shows us godel talks about true
and admits godel had no idea what true statement are
thus his theorem is meaningless
for if it read

Godel did is find a general method that enabled him to take any theory T
strong enough to capture a modest amount of basic arithmetic and
construct
a corresponding arithmetical sentence GT which encodes the claim ‘The
sentenceGT itself is unprovable in theory T’. So G T is true if and only
if T can’t prove it

If we can locate GT

, a Godel sentence for our favourite nicely ax-
iomatized theory of arithmetic T, and can argue that G T is
true-but-unprovable,

Godel did is find a general method that enabled him to take any theory T
strong enough to capture a modest amount of basic arithmetic and
construct
a corresponding arithmetical sentence GT which encodes the claim ‘The
sentenceGT itself is unprovable in theory T’. So G T is GIBBLY if and only
if T can’t prove it

If we can locate GT

, a Godel sentence for our favourite nicely ax-
iomatized theory of arithmetic T, and can argue that G T is
GIBBLY but-unprovable,

YUO WOULD SAY
HEY WHAT THE FU…K IS HE TALKING ABOUT WHAT MEANINGLESS DRIBBLE

you say

On top of that, saying that the argument “G is true though G is not in A” is “meaningless” is invalid; all we have to do is generalize our notion of “truth”.

what if you said

On top of that, saying that the argument “G is GIBBLY though G is not in A” is “meaningless” is invalid; all we have to do is generalize our notion of "GIBBLY.

without telling us what gibbly is
we would say
HEY WHAT THE FU…K IS HE TALKING ABOUT WHAT MEANINGLESS DRIBBLE

[denali]

You didn’t address my main point. Again.

The point of my second comment was that truth is either a) something we intuit and therefore talk about loosely, or b) a set of phonemes which we assign particular dialectical properties to, in which case truth can be a ham sandwich but the only difference is that n o o n e w i l l c a r e.

And no, I wouldn’t be speaking meaningless babble and no one with a functioning neocortex ought to say that I was any more than someone who comes up to me and starts speaking Quechua is speaking meaningless babble simply because I don’t speak Quechua and couldn’t recognize it as a language anywho. Information will still be encoded. Simply because language is opaque to you doesn’t mean there isn’t a meaning behind a word.