Hi, I'm just watching a lecture series and the professor writes on the board the following:
>Any axiomatic system that is powerful enough to encode the elementary arithmetic of natural numbers is either inconsistent or contains a proposition that can neither be proven nor disproven.
We'd be inclined to think that it is inconsistent because we want every axiom and proposition to be true in the axiomatic system if it weren't for the fact that proposition logic itself is consistent.
Since we know that propositional logic is consistent, meaning that you cannot prove every statement (eg. ((P and not P is false) is true)), wouldn't that make you inclined to go back and say that maybe the ex falso quodlibet law of logical systems might be a bit of a stretch in saying that the entirety of a consistent contradictory statement is true should remain in practice despite it not being provable?
If you cannot prove a premise, why would you even use it as a postulate/premise in a logical system?
>>8000910
I guess another way of reaching out to you guys is by asking:
Why would it not suffice to just say (P and not P "implies" false), rather than summarizing the entire statement as true and using that as a proposition?
I don't really understand your sentences, because they are unecessarily long, you use "it" without disclaimer, syrange quotation marks, I don't know if False and True is supposed to be part of your formal language, if you really want to ask about propositional and not predicate logic or how your question relates to incompletness.
Let me just point out that
1. there is paraconsostent logic (google it) which weakens explosion
2. an inconsistency such as 0=1 in arithmetic fucks you in a very practical way as 0+n=n
3. When in comes to classically interpreted propositions, intuitionistic (=constructive and therefore pretty well justified) logic has the same proof strengeth as classical logic (that had the axiom/law of excluded middle).
Why should I be inclined to think it's incosistent? It's simply that you introduce the axioms in a non secure way so you can't be sure you don't demonstrate the false. I don'd see what you're trying to say.
>>8000929
Sorry, I had
>and be put into practice despite it not being provable?
before instead of
>should remain in practice despite it not being provable?
which would tell you that "it" refers to the ex falso quodlibet law of logical systems instead of repeating it again. Language is very important, and I should proofread. Thank you.
If you can say that a false and a true statement is false and that that statement is true, can't you do that to all combinations and say that the statements are all true? Why would the ex falso quodlibet law be used if you can't do this to the rest? Another way of asking is how can you say that you can prove or disprove anything if you have two contradictory propositions in your axiomatic system? To deduce from ("All lemons are yellow" and "Not all lemons are yellow) that "Santa Claus exists."?
>>8000932
You need to know the definition of "inconsistent" first.
An "inconsistent" axiomatic system is one that contains a proposition that cannot be proven from the axioms.
>>8000950
That's incomplete. Inconsistent means that there's some statement such that you can prove both that statement and its negation.
>>8000942
Well have you looked into a reference for paraconsistent logics? Indeed, some people don't want to use an aciom that makes you throw away your system as soon as you can proof (A and not(A)) and they try and see whatever you may do instead.
For arthmetic however, as I point out proving two numbers like 0 and 1 being equal makes
5=5+0+0=5+1+1=7
provable, and all in fact every term in N is then equal to each other.
PS those logic don't have "3=3 is true" provable, they just have "3=3".
Gödels result is also completely syntactic, it's about provability, not truth as a mathematical entity.
You may look up Tarskis incompletness result to see that there can't actually be a truth evaluation in strong enough systems.
>>8000954
It occurred to me that by writing out my alteration to your definition, I was able to understand what you meant.
Very good. Thank you! I'm starting to realize that the only way I can understand something fully is by writing out my thoughts in clear form.
>>8000950
Watch your language on several ends.
A propositon is a statement that can formally be made, e.g.
3=3
3=4
2<1
The last two have not been proven in Peano arithmetic, i.e if the theory is consistent, then they are not theorems.
Peano arithmetic proves
not(1=2)
and we say it's inconsistent if it e.g. proves, simultanously to the above,
1=2.
Gentzen has proven Peano arithmetic proven consistent, though, btw., in a theory that axiomizes some large ordinals (but can't express all arihmetic "truths"). However, that theory has a domain of discourse too large to be comfy to some people.
Is less required axioms necessarily "better"?
Is having many axioms always a fault in a system?
>>8000965
In case anyone is interested, the following attempt to argue (>>8000954; him) helped me to understand my problem.
>The professor writes on the board
>>An axiomatic system is consistent if there exists >a proposition q which cannot be proven from the >axioms
>which would mean that inconsistent (its negation >or opposite) means
>>An axiomatic system is inconsistent if there does >not exist a proposition q which cannot be proven >from the axioms.
>Would this not mean that an axiomatic system in which all propositions can be proven is inconsistent?
Yes, it does. And since you cannot prove a statement and its negation [is this what is said?], then it is not inconsistent?
> Robert K. Meyer (1976) seems to have been the first to think of an
> inconsistent arithmetical theory. At this point, he was more
> interested in the fate of a consistent theory, his relevant arithmetic
> R#. There proved to be a whole class of inconsistent arithmetical
> theories; see Meyer & Mortensen 1984, for example. In a parallel with
> the above remarks on rehabilitating logicism, Meyer argued that these
> arithmetical theories provide the basis for a revived Hilbert Program.
> Hilbert's program was widely held to have been seriously damaged by
> Gödel's Second Incompleteness Theorem, according to which the
> consistency of arithmetic was unprovable within arithmetic itself. But
> a consequence of Meyer's construction was that within his arithmetic
> R# it was demonstrable by simple finitary means that whatever
> contradictions there might happen to be, they could not adversely
> affect any numerical calculations. Hence Hilbert's goal of
> conclusively demonstrating that mathematics is trouble-free proves
> largely achievable.
>
> The arithmetical models used by Meyer-Mortensen later proved to allow
> inconsistent representation of the truth predicate. They also permit
> representation of structures beyond natural number arithmetic, such as
> rings and fields, including their order properties. Recently, these
> inconsistent arithmetical models have been completely characterised by
> Graham Priest; that is, Priest showed that all such models take a
> certain general form. See Priest 1997 and 2000. Strictly speaking,
> Priest went a little too far in including “clique models”. This was
> corrected by Paris and Pathmanathan (2006), and extended into the
> infinite by Paris and Sirokfskich (2008).
>
http://plato.stanford.edu/entries/mathematics-inconsistent/#Ari
Something and its negation being false cannot be proven? Isn't that the most intuitively obvious statement? Of course it's false.... (Everyone in New York is gay) and (Not everyone in New York is gay) is false because either everyone in New York is gay or not...
I am so lost.
>>8000975
From your idiosynchatic way of > it's not clear when you're quoting and when you believe a statement or when you don't.
Good if the "That's incomplete" guy cleared things up to you, but be aware that your post he quoted surely didn't characterize incompleteness very well.
>>8000991
Something happened with the > and I don't know what it was. Might have had something to do with copying and pasting or something.
Everything ties in together well from his response it seems.
>>8000990
I think you still have to "prove" it though using axiomatic systems. They are saying that you can't, however intuitive it might be.
Please correct me if I'm wrong.
>>8000993
>Everything ties in together well from his response it seems.
...he said, before discovering an inconsistency
:)
>>8000990
That's a natural principle to adopt.
The issue is that naively writing down axioms might make you cook up a theory where this principle later turns out to be violated. (See what happened to Naivr Set Theory in 1901)
>>8000990
>Something and its negation being false cannot be proven? Isn't that the most intuitively obvious statement?
there is nothing trivial with the principle of explosion
https://en.wikipedia.org/wiki/Principle_of_explosion
and what you call false, bottom, 0 >has nothing< to do with truth. Logician tack the word truth on their formalization of deductive logic, but using the word truth (and truth value) is a personal interpretation of the formalization.
just learn formal logic
>>8001004
I don't understand the principle of explosion. How can you say that anything follows from two contradictory statements? Because in a world where a statement and its negation is true, our entire conception of reality crumbles and everything defaults to true into some fantasy world of irrationality?
>>8001001
>The issue is that naively writing down axioms might make you cook up a theory where this principle later turns out to be violated.
Which principle? That something and its negation being false cannot be proven or the principle of assuming "of course it's false because either everyone in New York is gay or not"?
Consequentia mirabilis was a pattern of argument popular in 17th century Europe that first appeared in a fragment of Aristotle's Protrepticus: "If we ought to philosophise, then we ought to philosophise; and if we ought not to philosophise, then we ought to philosophise (i.e. in order to justify this view); in any case, therefore, we ought to philosophise."[3]
The most famous example is perhaps the Cartesian cogito ergo sum: Even if one can question the validity of the thinking, no one can deny that they are thinking.
>>8001015
>Because in a world where a statement and its negation is true, our entire conception of reality crumbles and everything defaults to true into some fantasy world of irrationality?
you think this only because you have been trained to think like this.
People just love to seek a structure for their favorite speculations, and a few people love to formalize their mental proliferations.
>>8001017
>if we ought not to philosophise, then we ought to philosophise
But you just said you ought not to philosophise... you can say from that you ought to philsophise?
>>8001017
>Even if one can question the validity of the thinking, no one can deny that they are thinking.
I guess I'm stupid because I don't see how that's related at all. :(
>>8001016
The former. E.g. see Russels paradox for a theory of sets.
Theories about gay peopld are hardly ever formalized, although if you read up on e.g. Marxism, those people write more formal than many other non-stem texts (they have axioms, but they still step out the formal realm on every other occations, so it's worth nothing form a mathematicans perspective)
buopm
