Does there exist a deductive system for which there are no undefined terms?

Sure? IIRC, a point is defined as a dimensionless entity.

They don’t need to be defined. Or rather, any 2-value logic systems can be established without these.

Points are undefined in geometry.

Urban Ranger: Points lack a formal definition. Other things are formally defined in terms of points. If points had to be formally defined as well, they’d need something to be defined in terms of, too. Then how would that be formally defined? If you choose to define something in terms of that which it formally defines, you’ve created an inconsistency in your logic and your system collapses. Therefore, you must have at least one (or, in the border case of an empty logical system, zero) thing that is simply introduced, but not formally defined.

“Line” and “plane” are also undefined in geometry. Euclid’s first couple of axioms used those terms without definition.

2-value logic means logic that uses the states “true” and “false”. Necessarily, they are not defined. There are other logics, though. There is even 5-value logic that uses the states “true”, “false”, “true-false”, “false-true”, and “glitch”.

(BTW, thanks for the link, Lib.)

Am I right in thinking that the empty set doesn’t count as undefined - an axiom asserts its existance, but not explicitly, just by saying there exists a set no other set ‘is a member of’.

Can we not manage to shift all definitions to axioms in this way? Define a language a lot like logic, but have a,b,c… as object variables, and A, B, C… as relation variables. (Use E for ‘exists’, and A for ‘for all’) So for instance sentances could be ‘Ex’ or ‘**EA:A(x,x)^A(x,y)=>x=y’. Have I mucked up the definitions here, or is this as good as logic?

Then we can define ‘member of’ by:

EE
AE,F:E=F

Sorry for the incoherance, but do you see what I’m saying?

The empty set may or may not be defined depending on the deductive system that uses it. If it is a “primitive”, i.e., one of the initial building blocks of the system, then it won’t be (and in fact, can’t be) defined.

It’s important to understand that just because a statement contains the word “is” (or a symbolic equivalent), that does not mean the statement is necessarily a definition. For example, this statement is not a definition: “George is fifty years old”.

Here’s one with symbology:

[symbol]"[/symbol]x[symbol]$[/symbol]y(y=x)

Although at first glance, that might look like the definition of necessary existence, it is in fact an assertion that necessary existence exists. You might recognize it as the so-called “Modal Axiom”.

Sorry, I didn’t have any reference materials to hand when I wrote my last post. To clarify: Consider ZF set theory as described here http://en.wikipedia.org/wiki/Axiom_of_empty_set . Is the ‘empty set’ undefined? I would guess not, as there’s no explicit mention, just an axiom that says ‘Ex:Ay:(y not in x)’. (It is refered to as well in the axiom of infinity, but I think that’s just shorthand for something like the empty set axiom.)

Typically — and I believe this includes ZFC (which is ZF with the Axiom of Choice included) — set theory begins with three undefined concepts: [1] equality, [2] membership, and [3] ordered pair. Here is some explanation. The empty set is an axiom of the Zermelo-Fraenkel system. (It also postulates an infinite set.)

Cool.

So, given that, how about my attempt at a no-undefineds system? Actually, on re-reading, I think I used ‘=’ without definition. But you see what I’m getting at, right? Can we not have a language where we can use axioms to define the existance of whatever relations we need, in the same way we use an axiom to assert the existance of an empty set?

Well, definitions and axioms are fundamentally different conceptually. Deductively, you can’t just define something into existence. Proof requires a rigorous chain of valid inference that begins with axioms and ends with a conclusion. If all the inferences are true, then the proof is “sound”.

Otherwise, I could define God into existence by defining God as E. That is indeed an appropriate definition, but it proves nothing. In the same way, I could define 2 as 1 + 1, but that would not prove that 1 + 1 = 2. Peano stated 5 axioms before concluding that 1 + 1 = 2.

Again, there is a root problem that is epistemological in nature. The moment you begin to use terms (or symbols) you must either leave them undefined or define them. And if you define them, then the terms (or symbols) in your defintion must be left undefined or else be defined. And the cycle continues until you end up with terms (or symbols) defining one another in a circular logic hell.

I thought this was what you were asking us?

Yes, but you can state some things as one or the other, yes? You can say ‘Exists x s.t. no y is in x’ or ‘no y is in phi’, which produce a similar concept - an empty set - but one is an axiom and one a definition. Is that right?

Anyway, I’ve been thinking about what I posted. I think that given some framework to work in, you could get a system with no undefined terms, but that that must have undefined terms.

Look at Hilbert’s axioms on: http://www.math.ohiou.edu/~connor/geometry/chap3/betweenlct.html

You might be able to avoid having undefined ‘point’ and ‘line’ but then someone would point to ‘set’ and ‘contained in’

Well, Hilbert’s axioms actually describe an important modern system of deduction in which the modus ponens is the only rule of deduction: p -> q, p :: q. But there are certain shortcomings with the Hilbert system, especially in algorithmic problems and other logic spaces where modus tollens and other rules are required.

With respect to definitions and axioms, a definition is descriptive and an axiom is assertive. As I explained before, no inference can be drawn from a definition because no implication is made with it. Another way to look at it is this: before the statement of axioms is underway, the statement of all definitions must be completed. The definitions contextualize what it is that the axioms will assert.

As for what I was asking, that has been answered by Ultrafilter. As I said before, the question has been asked and answered.

OK, thanks.

Gödel’s Incompleteness Theorem has a summary of the main points to the issues included in Godel’s theorem, for any who may be interested.

The difference between a definition and axiom is this: you gain nothing new by introducing a new definition, but possibly quite a bit by introducing a new axiom.

Every statement in arithmetic may be expressed in terms of 0, ', and =. There’s no need for symbols like 3, +, and <, but they serve to make proofs quite a bit shorter.

I think that fits nicely with how I described it as well: a definition is descriptive and an axiom is assertive.