Although, beyond any doubts, $ZFC$ is by and large the predominantly accepted theory of sets, there have been a few attempt to establish some serious competitors in town. 


I just quote two of them (there are several more): $NF$ by Quine and Alternative Set Theory by Petr Vopenka. I think those attempts are epistemologically interesting, in that they open doors to quite different views about the world of sets and how we conceptualize them (and therefore on the entire cathedral of mathematics grounded in set theory). 


Now, here is my question: is there something like it in formal arithmetics?


Are there Alternative Formal Arithmetical theories? 


I do NOT mean the various fragments of arithmetics, which essentially start from Robinson Arithmetics $Q$ (or even Pressburger's Arithmetics) and then consider some limitation of the infamous Induction Rule (IOpen, $I\Delta\_0$, $I\Sigma\_n$, etc.). All those share the common denominator $N$, and of course they differ in the "nonstandard models", as well as their proof theoretical strength.


I mean some formal systems of numbers which substantially move away from the traditional picture of $N$, all the while retaining some basic intuition of counting, ordering, arithmetical operations. 


To give you an idea of what I am after: systems in which it is not true that all numbers have a successor, or it is not always true that $Sn\succ n$, or one in which the ordering of natural numbers is not linear or even not total, or an arithmetical first order theory whose intended model are the countable ordinals. 


Or perhaps even wildest animals.



Alternative Arithmetics
Recall that $NFU$ is the Quine-Jensen system of set theory with a universal set; it is based on weakening the extensionality axiom of Quine's $NF$ so as to allow *urelements*. 


Let $NFU^-$ be $NFU$ plus "every set is finite". As shown by Jensen (1969), $NFU^-$ is consistent relative to $PA$ (Peano arithmetic). $NFU^-$ provides a radically different "picture" of finite sets and numbers, since there is a universal set and therefore a last finite cardinal number in this theory. 


The following summarizes our current knowedge of $NFU^-$.


**1.** [Solovay, unpublished]. $NFU^-$and $EFA$ (exponential function arithmetic) are equiconsistent. Moreover, this equiconsistency can be vertified in $SEFA$ (*superexponential* function arithmetic), but $EFA$ cannot verify that Con($EFA$) implies Con($NFU^-$). It *can* verify the other half of the equiconsistency.


**2.** [Joint result of Solovay and myself]. $PA$ is equiconsistent with the strengthening of $NFU^-$ obtained by adding the statement that expresses "every Cantorian set is strongly Cantorian". Again, this equiconsistency can be verified in $SEFA$, but not in $EFA$.


**3.** [My result]. There is a "natural" extension of $NFU^-$ that is equiconistent with second order arithmetic $\sf Z\_2$.


For more detail and references, you can consult the following paper:


A. Enayat. *From Bounded Arithmetic to Second Order Arithmetic via Automorphisms*, in **Logic in Tehran**, Lecture Notes in Logic, vol. 26, Association for Symbolic Logic, 2006.


A preprint can be found [here](http://academic2.american.edu/~enayat/Enayat-Tehran-August%252024.pdf).


