My understanding has always been that an axiomatic system requires certain undefined terms, called primitives, upon which the system’s defined terms and axioms are built. The point, obviously, is to avoid circular definitions.
For arithmetic, “successor” is undefined. (Successor is a term used to state the Induction Axiom). For geometry, “point” is undefined. And for systems of 2-logic, both “true” and “false” are undefined.
Does there in fact exist a deductive system that has no undefined terms and no circular definitions?
Godel’s proof has nothing to do with that. And every deductive system must have undefined terms.
Hmm…
It’s possible that if you allowed infinite-length definitions, you might be able to get away with defining all your terms. But that’s just gonna get ugly, and it’s not clear what you would gain by taking that route.
Think of it this way: we can represent the definitions of the theory as a graph, with the vertices corresponding to the terms, and an edge from v[sub]1[/sub] to v[sub]2[/sub] iff the term corresponding to v[sub]2[/sub] appears in the definition of v[sub]1[/sub]. This won’t generally be a tree (as you won’t be able to find a unique root), but if you want to avoid circularity in your definitions, you will end up with a directed acyclic graph (DAG). Every finite DAG has at least one vertex of outdegree 0, and that will correspond to an undefined term.
Hmm. Would you consider the “MU” system from Godel, Escher, Bach to be an example of what you’re looking for, or not? I don’t remember the rules exactly, but that’s a system with axioms and rules of inference where the symbols are completely meaning-free: it’s just a game for manipulating strings of letters. (As I recall, the sole axiom was “MI”.)
If a logic has no terms, the empty set is a model. I suppose a logic could have only terms defined in terms of themselves or of each other. But that would also have the empty model.
There is the empty logic. No terms and no axioms. It has a unique interpretation. It is even complete, since every statement is provable. (There are no statements, since no terms.) Not very interesting.
Eh, Libertarian’s just trying to get away with claiming that “goodness is that which God values” without explaining what God is or precisely what things it values.
A point is just an element in geometry: a “thing” that acts as the basic unit. There’s really no way to define a point in geometry, but that isn’t really a problem: what do we mean by “point” in real life?
OK, to bring something out of the possible tailspin this thread has gone into: You could, it seems, define a logical system with the terms good' and God’ undefined (that is, created as primitives) and the concept of `value’ introduced via an axiom (that is, created as a correct statement that is unprovable within the logical schema). Such a logical system is conceivable.
But what good would it do us? What natural processes would it describe? We can use the calculus as a schema out of which we construct descriptive models of certain behavior (for example, the velocity of a falling object as time varies). What would the `good-God’ axiomatic system do?
A reasonable question, Derleth. (And a wonderful pun!)
For me, the system is important in helping me to understand my purpose in life. I therefore find it useful, and it therefore does some good for me. It likewise could do some good for anyone who, upon accepting its premises, finds purpose where before he found none.
At any rate, I do believe the question has been asked and answered. There does not exist a deductive system for which there are no undefined terms.
Here is one of many references (Google: peano successor undefined). Undefined terms are simply the starting point for deductive systems. Without them, there would be circularity.
A clarification please. So there are three things which might qualify for the OP (circular reasoning, infinite regression, empty set)?
Godel’s proof was the first thing I thought of when I read the title. I thought it would be a conclusive answer to the OP. Is there something I’m missing or did I not read your post correctly?
No system can define its most basic elements – that much is obvious.
But those elements can still be defined in more complex systems. For example, “ordered succession” might not be defined in arithmetic, but it’s trivially easy to define in real life: we take a number, apply a particular operator to it, and the number we get back is the number that “comes after” the first.
Right. If you take the axioms of natural numbers as your starting point, 0 and x’ (the successor of x) are undefined. If you take the axioms of ZFC as your starting point, you can define the natural numbers in terms of sets, and then 0 and x’ are defined. But you run into undefined terms in set theory (just [symbol]Î[/symbol], IIRC).
The problem is that there are a finite number of words.
Therefore, definitions are necessarily circular by nature. Think of it this way: you have a Term and its Definition, which consists of other Terms. Each of those Terms has its own Definition, and each of those Terms in each of those Definitions have their own Definitions with their own Terms.
At some point, you’re going to have to use a word to define a word that has defined the word you’re using.
Ah, but deductive systems can accurately model any number of processes. Therefore, since Godel’s Incompleteness Theorems limit those systems, they limit the processes as well.