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Title says it all.

My understanding is that axiom system is a set of true propositions which are promises and self-evident. And to prove a proposition, we use deductive way to abbreviate as axioms or axiom based proved theorems or lemmas. From this view, I thought I could think true as axioms itself.

However, while I'm studying mathematical logic, the text book states true without introducing any axioms, but some definitions. They states; for every rule of the system, the following holds: whenever the elements are derivable in a system, they have the property $P$ which holds for every elements derived from a system. (Property $P$ defined as a function $P: X \rightarrow\{\text{true, false}\}$ where $X$ is a set of all strings. Iff the value is $\text{true}$, we say the element has the property $P$ in a system.)

Is this the definition of mathematical true and falsity?
If it's not, how to rigorously define true?

Shin Kim
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1 Answers1

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It is not so easy to assert that all axioms of all mathematical theories are "self-evident".

For sure, in mathematics, like all other sciences, "first principles" (i.e.axioms) are assumed to be true (until revision).

Logical inference rules are "designed" to be sound: i.e. to "transfer" truth.

Thus, deductive consequences of our axioms will be true, provided that the axioms are.

Added

If I understand your proposal, after the comments, you start from the set $Z$ of expressions (i.e.well-formed strings of symbols of the language) "generated" with a system; if we interpret system as a "typical" couple :

axioms + inference rules,

the set $Z$ of all expressions "generated" with it is what we usually call the set of theorems.

Now you introduce the characteristic function of $Z$, i.e.:

$P_Z(x)=1$ iff $x \in Z$.

Thus, your proposal is to "define" :

$T(x)$ [i.e. $x$ is true] iff $P_Z(x)=1$.

Assuming that my interpretation is correct, we have a couple of problems :

(i) Gödel's Incompleteness Theorem applies to "significant" formal systems, including most of "interesting" mathematical ones :

in any consistent formal system $F$ within which a certain amount of arithmetic can be carried out, there are statements of the language of $F$ which can neither be proved nor disproved in $F$.

Thus, according to our "natural" insights regarding truth, we have mathematical truths expressible in the system $F$ which are not provable in it, i.e.are the class of theorems of the system $F$ does not "exhaust" all the "true facts" of the system.

(ii) if your system is inconsistent it will "generate" as theorem all well-formed "expressions" of the language. Thus, the set $Z$ of theorems will be "too large".

Mauro ALLEGRANZA
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  • I appreciate your answer and also perfectly agree with this. But it's not suitable answer for my question. I'm asking "is my understanding correct?" or "if it's not, how to rigorously define true?". – Shin Kim Apr 12 '14 at 11:19
  • @ShinKim - Form Plato and Aristotle we are still struggling with the "definition" of truth; this, I simply do not think that you can have a "scientific" definition of it. About math logic, the usual def consider a valuation $v : X \rightarrow { true, false }$, where $X$ is the set of atomic formulae. Then the usual semantic rules for connectives and quantifiers propagetes this definition to all well-formed formulas. Lastly, inference rules are designed in order to be sound, i.e.to derive true conclusions from true premises. Thus, if we start from true axioms ... 1/2 – Mauro ALLEGRANZA Apr 12 '14 at 11:24
  • ... all theorems derived from them through correct applications of the infernce rules, are true either. 2/2 – Mauro ALLEGRANZA Apr 12 '14 at 11:25
  • Let $\mathcal{A}^$ denotes a set all strings, and let $Z$ be a set whose elements were generated by means of a system $\mathfrak{C}$. We define property function $P_\mathfrak{C}:\mathcal{A}^\rightarrow{\text{true,false}}$ iff $P_\mathfrak{C}(\zeta)=\text{true}$ where $\zeta\in Z$. Isn't this a "scientific definition"? – Shin Kim Apr 12 '14 at 11:38