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Recently one of my friend and I are working on a project on a certain generalization of set theory. This is the same friend of mine of whom I talked in this post. However, the basic outline of his (modified) theory is as follows,

As usual, the logical symbols of our alphabet are,

  • the quantifiers $\exists$ and $\forall$.

  • the logical connectives $\land,\lor,\iff,\implies$ and $\neg $.

  • Parentheses, brackets, and other punctuation symbols.

  • An infinite collection of variables which intuitively represents what we call abstract objects, a sub-collection of which is divided further into three sub-collections, namely mathematical objects, ordered objects and relational objects. We denote the variables representing mathematical objects by $a,b,c,A,B,C,\ldots$, the variables representing ordered objects by $\alpha,\beta,\gamma,\Phi,\Psi,\Theta,\ldots$, the variables representing relational objects by $\mathfrak{a,b,c,A,B,C,\ldots}$

  • An equality symbol (sometimes, identity symbol) $=$.

and the non-logical symbols of our alphabet are,

  • A predicate symbol $\color{crimson}{\text{P}}$ (or relation symbol) with number of arguments $2$.

Now we say that a relational objects $\mathfrak{R}$ is definable from $a$ to $b$ if, $$(\color{crimson}{\text{P}}ab)\land(\exists\gamma)[\color{crimson}{\text{P}}\gamma a\land \color{crimson}{\text{P}}\gamma b\land (\mathfrak{R}=\gamma)]$$

My question is can we do that? I have a feeling that my friend is missing something but since I am not so expert in formal logic, I can't say anything precisely on this matter.

Can anyone help?

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  • The sentence "Now we say $\dots$" is apparently intended as a definition of a ternary predicate called "definable", a relation between $\mathfrak R$, $a$, and $b$. I don't think "definable" is good terminology for this predicate, since this word already has a standard meaning in logic. Apart from that, the proposed definition is permissible (because definitions are largely arbitrary) but it seems quite useless. Its significance, if any, will depend on the axioms, not mentioned in your question, that your friend assumes about the primitive notions of this theory. – Andreas Blass Jul 23 '16 at 05:05
  • also not mentioned are the rules for making well-formed formulas and the inference rules. For example, for now I have no idea how a greek letter is different from a latin letter or a fraktur letter. Also your formula looks weird because if $\gamma$ exists it has to be $\mathfrak R$ so you can just replace $\gamma$ with it and remove the quantifier. – mercio Jul 23 '16 at 05:09
  • @AndreasBlass: Yes you are right. The sentence "Now we say ……" is actually intended as a definition of a ternary predicate called "definable" already mentioned in my earlier post. By the way, since you have answered my question (I think so), can you consider it posting as an answer so that it doesn't become listed in the list of unanswered questions? –  Jul 23 '16 at 05:10
  • @mercio: One explanation for this (as I have understood by talking with him) is that the only difference that we are allowing between say, $\gamma$ and $\mathfrak{R}$ is in the way we denote it. In other words, the difference is in the way we "see" them. If we see the object as an ordered object then we use greek letter to denote it and if we see the object as a relational object then we use fraktur letter to denote it, although they may be identical. –  Jul 23 '16 at 05:16

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If your friend is attempting to construct a usable formal system, then it is necessarily one that can be implemented by a program. You need a program that given any input strings $p,x$ will always halt and output whether $p$ is a valid proof of $x$ (representing a theorem) or not. If you cannot see how this can be done (at least in theory), then it is unlikely that you have described any usable formal system at all.

As I suggested in my answer to your previous question, you should study logic and existing formal systems first before you attempt to build your own. Little of your question as stated makes sense, precisely because you're not precise enough:

  • You say there are different kinds of objects. In logic that is called "sorts", and you need to specify clearly how various sorts interact.

  • What are the inference rules for "P"?

  • Is "definable" an internal symbol in your system or not?

It is not sufficient to be able to answer these questions in an intuitive way; you have to describe purely syntactical deterministic procedures, that can be unambiguously executed by anyone who can follow instructions on paper to manipulate strings, and will result in a definite conclusion to whether or not a purported proof $p$ of a purported theorem $x$ is valid or not. There must be no reliance at all on intuition or any mathematical knowledge.

Only after you have pinned down the syntax of your formal system in this way is it then possible for us to talk about it. In particular, we would then be interested in its semantics, namely whether or not there is a reasonable interpretation of the provable theorems. This is a completely separate question from that of syntactic consistency. For example in first-order logic a theory is said to be consistent iff it cannot prove a sentence of the (syntactic) form "$P \land \neg P$". As I mentioned in the post I linked to from my other answer, this only requires a weak meta-system to define. But if you have a strong enough meta-system you can further talk about models of a theory and can prove that consistency implies the existence of a model.

Yet consistency is not going to be enough if you desire a formal system that is foundational in any sense. Note that $PA + \neg Con(PA)$ is consistent but proves itself inconsistent, so it is simply useless for a foundation for mathematics or philosophy. What at least you would need for a system to be justifiable as foundational is that it has an ω-model, which at implies both arithmetical soundness and ω-consistency.

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