Logic question - finiteness definable?

In my undergrad logic course, we came across a couple of results about finiteness not being definable, using compactness of 1st order logic.  In particular, I remember the class of finite sets not being axiomatizable (in the sense that there's no set of wffs S such that A |= S iff A is a finite structure).

This runs contrary to what I've seen in discussions of set theory, i.e. defining a subset of A as {x in A : x is finite}.  But can we construct a first order wff P(x) meaning "x is finite"?  I guess I'm just asking for a formal justification of definitions like this.  Any help would be appreciated.

Something along the lines of: A set S is finite if there exists some function N where N(x[size=9]i[/size]) = x[size=9]i+1[/size] and There exists some element E such that for all other e in S e < E.

Would that work?

>>2

Also, I fail at bbcode, wooooo!

>>2
what is this relation <?  How does S having an <-maximal element E make it finite?  And is N:S->S?  It assigns x_i to x_{i+1}; what's the index i, a natural number?  We are assuming S is countable then??  I'm sorry, I really don't know what you're going for.

But what about a wff P(X), meaning "There exists an injective function mapping X to a proper subset of X".  This seems pretty easy to formalize; then not(P(X)) means "X is finite".

Then still, how to reconcile this with the logical results mentioned (admittedly vaguely) above?  Any logicfags help?

Hmm, not logic-fag but I'll try anyway.

Let's use a language L=(in, f) where in is a binary relation symbol and f is a unary function symbol (the rest of FOL-language is implicit, hope you understand my notation). In addition I assume we  need a collection S of axioms for set theory.

So what about:

A(X):= for all (x in X):  (f(x) in z)  [f:X->X]
B(X):= for all (x,y in X): ((f(x) = f(y)) -> (x=y)) [f is injective]
C(X):= exists (x in X): ( for all (y in z): (f(y) != x))[f(X) is a proper subset]
D(X):= A(X)&B(X)&C(X) [X is finite (?)]

Can we find a model for S and the sentence 'exists X: D(X)' ? If S = ZF?

(Waiting for logician in shining armor to clear up mess)

I'm not sure how to put this, but I'll try to keep my statement to a minimum. Wish me luck.
Ok...The things that are finite in our universe are defined by the contributing and segregative factors towards a harmony of balance. ie acid and bases are what make up one of our harmonies that we must contribute to ourselves in order to stay alive. So, if one of those factors should overwhelm the other, eventually we will see a slow degredation of solidity (chemical-molecular breakdown) or a slow solidifying effect (petrification). We have seen these effects for years, but are still not so familiar with what causes the breakdowns to occur. What is infinite is the contributing and segregating factors and nature (meaning to return to a state of subtle harmony) and lex talionis (meaning retributive factors) and Action|Reaction (+A1- = +Ra1- + -Ra2+) for every action (+A1-) there is an equal reaction (+Ra1-) and oppossing reaction (-Ra2+) where reaction (-Ra2+) opposses both (+Ra1- and +A1-) which are contributive factors.
Example, for social studies, there is a dictator who performs 1 action that segregates (+A1-) meaning that it causes conflict between combined parties (+Ra1-) the retributive factor would then be (-Ra2+) meaning there would be a unification performed at some point to oppose the dictator and similar activities, thus the nature of the combined parties change to that of active (+A1-) and the dictator becomes that of reactive (+Ra1-) and the retributive factor would be the outcome (-Ra2+). So, what happens if (+A1-) performs an action that segregates self from community (+Ra1-)? It would then lead to seperation, and finally there would be a reaction of other natures to re-combine with the person that seperated (-Ra2+). And the cycle continues for infinity. To argue whether time is finite, all you have to do is ask yourselves if we as a human species don't exist what IS time? Time will cease (+Ra1-) only when motion ceases (+A1-), the retributive factor will be a re-activation of motion (-Ra2+). All of this is based off of science fact, but it still is an axiom due to be not being ABSOLUTELY sure that everything I say is true, it's just a best guess at this point. But, with acceptance of this and exercise (+A1-) the result will be either a resounding false (+Ra1-) or a resounding truth (+Ra1-), when (+Ra1-) is identified the oppossing factor will be easily recognizable as (-Ra2+). So, what best guesses would any of you have if (+Ra1-) is truth? Then, the (-Ra2+) would be the oppossing factor of "NO, YOU ARE WRONG." where the oppossing factor has yet to perform (+A1-) to discover whether (+Ra1-) is true or false, so (+Ra1-) remains in the realm of imagination due to lack of (+A1-) action. So, how would imagination work then? Hehehe, that's another axiom.

Ah jeez, I forgot to include the nature of things, crap, I'm such a sorry-ass scientist. :P
Ok, the reason for lex talionis is the retributive factor that returns the reaction back to its original state by (+A1-) the reaction is the bringing back of original nature, but that time always moves forward that places the occurrence further in the future or leaves the original nature further in the past, but this then becomes a recurring cycle of nature and thus, to repeat this becomes the nature depending upon contribution or segregation. Ok, that's all I wanted to say.

>>5
Looks right to me, except for the minor omission of "not"s in the statement of D(X);  and I think you've highlighted the distinction between the result I mentioned and what we're trying to formalize.  A model A of ZF is necessarily infinite, while a finite subset X in the model A should satisfy the wff D(X).  And I'm not sure if the function symbol is necessary, since we can talk about functions as sets of ordered pairs within the language of set theory.

On the other hand, our result from class stated that no set of wffs exists such that a structure is finite iff it satisfies the wffs.  That is, regardless of what relation/function symbols the structure is endowed with.  This is a somewhat different statement than saying that finite sets are not definable within ZF.

Also, when I talked about finiteness not being definable, I may have been thinking of the finite elements in the nonstandard model of arithmetic.  My apologies, for I am still very confused by all this :/

To clean up the first para-para; I'll bring you into the enlightenment of the nature via science.
1 If I, (A2U), perform (+A1-)1 and discover (+Ra1-)1 = true goto line 2
2 If true, then (A2U) performs speech to make aware of (+A1-)1 by speech, then (-Ra2+)2 would be more prominent than (+Ra1)2 meaning no one would believe me at first, but upon them performing their, (PEOPLE), (+A1-) with accepting my original statement as an axiom for them and a paradox for me, where (A2U paradox exists in memory as experience) and (PEOPLE axiom exists in memory as imagination) they would then discover for themselves what is true and what is not. What do you think? Am I a jackass for making such obscure and ambigiuous statements? I'm a quack right? Well, what do you think they said to Benjamin Franklin when he said he was looking for something like electricity? How about Thomas Edison? I bet everyone around him was telling him that making a lightbulb was a complete waste of time, yet he continued on for over a thousand tries ending in complete failure, but he only needed one way for the lightbulb to work, this is the harmony of balance of dualities. This harmony, continually changing throughout time. So, I am already working on a time-line factor of harmony of dualities and illustrating what change is, what causes change, what effects come from change, and what are the retributive factors that come from these? I state all of this not to presume my credibility of (RIGHTNESS), but to be a contributing factor for you (PEOPLE), to get off your asses and prove me wrong. Science is meant to insite action, not stop it. To stop an action by performing an action would mean that less are doing the less work for more. WHAT?!? XD

This original thread is asking about finite and infinite logics, and that is what my axiom of "cradle of life" highlights.
For every life there resides a cradle that the life resides within, thus this cradle is infinite only to the finite life which is created inside the cradle.
Example, if our human bodies are cradles, the bacteria, fungus and bacteria that live inside us will be born and die during the course of our life-times over and over again. So, to those bacteria we are infinite.
However, take into account that we are a life-form that is born and dies within the realm of space and planets and stars, so to us TIME is infinite due to continual existence of other entities. Basically what cradle are we within that appears infinite? Has anyone asked such a compelling question?
And this axiom is re-inforced by Albert Einstein's "Laws of Relativity"

>>1
If you can count it, it's finite.

presto, definition

>>11
Countable lol?

>>8
lol oops that 'not' was of course meant to be there. Also I see how I didn't really answer your question. I wouldn't be very surprised if finiteness is non-axiomatizable but I'd like to see a proof.

Given a number n it should be easy to make axioms so that any model for them has n elements, but it's a very different thing to axiomatize the existence of such a number in FOL.

>>11, WHOA! What a revelation! >>11, what is the last number in our number system, please do tell. :)

>>13, GROW UP! YOU DON'T GET PROOF! YOU EARN IT BY DOING THE WORK YOUR SELF! Jezus, what is with these people always want us to do the work for them when they won't even do it for themselves? IT'S VERY DISRESPECTFUL!

>>13
Here you go: suppose, for contradiction, that the class of finite sets is axiomatizable: let S be the set of wffs meaning "X is finite".  Then union S with all the wffs "X has size at least n" for each natural number n; call the union S_1.  S_1 is finitely satisfiable, obviously: in a finite subset S_0, there is some largest n that appears, so S_0 is satisfied by a model with n elements.  Then by compactness, S_1 is satisfiable.  But a model X for S_1 is clearly infinite, though we should have X satisfying S - contradiction.

Nice, thanks

>>16
Assumes the validity of induction.

>>16 and >>18, YES!!! Assuming something is true without proof only at the start is an axiom. What we do to test that axiom is called questioning, after the questioning comes the paradox where what we really weren't sure of the validity of the axiom, now has been proven and is now paradox. GOOD JOB!

>>19

Shut the fuck up

>>20

That's all the axiom I need.

>>21, Really? I guess that means you wouldn't make a great scientist then. You see, in order to be a scientist or mathematician you need axioms. So, good luck on being blissfully content.

>>20,
"Shut the fuck up" <Axiom>
>>AnOnYmOuS 2U
"Why should I 'Shut the fuck up'?" <Question>
"Who is going to make me 'Shut the fuck up'?" <Question>
"What will make me 'Shut the fuck up'?" <Question>
"When will I 'Shut the fuck up'?" <Question>
"How will I 'Shut the fuck up'?" <Question>
"Should I 'Shut the fuck up'?" <Question>
"Could I 'Shut the fuck up'?" <Question>
"Would I 'Shut the fuck up'?" <Question>

Congratulations, you just helped me prove scientific proof method. :)

someone is much elated with their discovery of the axiomatic proposition system
gj

>>18
Is that a serious problem?

Only to someone seeing the frog in a pot of water that's slowly being brought to a boil. :)

ETHEREAL

Swift shimmering shadow steps silently; saddened, still softly severing serenities, slicing sympathy staggering sturdily; suffering sarcastic solace. Never noticing naïve naiads nor narrows nor necrotic nethers nestled nearby; naughty neutral nocturnal nexus nevertheless normally nostalgic notions; noteworthy nemesis neo neurotic neuralgia. Exceptional effervescing elemental ephemeral entity experiencing euphoria everlasting eternally eschew evermore, Ethereal

Has someone answered this? I'm not going to check. Anyways, an interval [a, b] is of finite length if for all x, a < x < b. But the intermediate values are infinite. So it only holds if you are speaking of a definite transition from point a to point b. Otherwise you are gay and like it in the ass.

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