# Would this argument support logicism?

Notice the following theory written in multi-sorted first order logic with equality and membership, with axioms of:

Comprehension:$$\small k=1,2,..,\omega, \omega+1,..$$ $$\forall x^{i_1},...,x^{i_n} \exists x^k: \forall y^{

Sorts: $$for \ \small i < n, j\geq n$$ $$\forall x^i \exists x^{

sorts when not otherwise specified are taken to range over all metatheoretic ordinals $$\{0,1,2,..,\omega,\omega+1, \omega+2,..\}$$

This theory interprets Zermelo set theory + Rank; so it constructs a cumulative hierarchy of size $$V_{\omega + \omega}$$

The caveat here is that this is too near to logic! The sorting axiom can be considered as belonging to the underlying multi-sorted FOL with identity. We only have ONE set scheme that is comprehension. Not only that we can push this further into the realm of high order monadic logic with identity and accumulative sorting, i.e. higher sort unary predicates can take any lower sort unary predicates as their arguments, but the opposite is not allowed, so $$x^i(x^k)$$ is only a formula if $$k. Now the axioms would read as:

Comprehension:$$\small k=1,2,..,\omega, \omega+1,..$$ $$\forall x^{i_1},...,x^{i_n} \exists x^k: \forall y^{

Sorting: $$\small i < n; \ j \geq n,\ k>i, \ k>j$$

$$\forall x^i \exists x^{

here $$y^{ would be a variable that range over unary predicates of all sorts below $$k$$, while variables like $$x^k$$ denote variables ranging over all unary predicates of sort $$k$$.

We can push that further (go beyond metatheoretic ordinals less than $$\omega+\omega$$) as long as we have ordinal notations, i.e. as long as our metatheoretic ordinals (indexing sorts) are below $$\omega_1^ {CK}$$.

The point here is that the comprehension axiom can be understood as being about logic, and so a logical scheme.

It appears that $$\sf ZFC$$ and higher extensions, all can have models in sets belonging to these theories.

why that won't count as an argument in favor of logicism?

• Welcome to MathStackExchange Prof. Holmes! Yes! I smoggled infinity into the background sorting of logic, and I think this move is innert! I mean as far as the question of whether the theory is logic or not? Why should for example finite sorted logic be considered as logic while the infinite sorted not? One can easily symbolize the sorts using "pairs" of naturals so we can use $0,n$ notation to represent the finite sorts and the $1,n$ notation to represent the 'infinite' sorts, I don't think this matter by itself is related to logic or not logic question. That's my own stance on it Commented Jun 23, 2020 at 8:09