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As far as I can tell, there are certain non-stratified formulas $\phi(x)$ such that NF proves that $\{x : \phi(x)\}$ exists. For example, fix a set $y$, and let $A = \{x : y \in x\}$ and $B = \{x : x \in y\}$. Then the set $A \cup B = \{x : x \in A \vee x \in B\}$ exists. We have $A \cup B = \{x : x \in y \vee y \in x\}$, so the non-stratified formula $x \in y \vee y \in x$ defines a set.

Can anything be said in general about what classes of non-stratified formulas define sets in NF?

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    $\begingroup$ I'm no expert on the matter, but what exactly do you have in mind? As far as I am aware, the only non-stratified stuff you get in $NF(U)$ is anything you can generate out of objects you already have and Boolean operations. Thus if you can generate a particular set by a non-stratified formula, it must have come from a Boolean operation on something defined in a stratified manner. $\endgroup$ – Neil Barton Sep 18 '13 at 20:42
  • $\begingroup$ @NeilBarton: If anybody can confirm your idea, then that would be an answer to my question. I am specifically wondering if and to what extent non-stratified formulas involving quantifiers can be imitated. E.g., what about $\{x : \exists z ((x \in z \wedge z \in y) \vee (y \in z \wedge z \in x))\}$? $\endgroup$ – Nick Thomas Sep 21 '13 at 21:04
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In $NF$ non-stratified formulas can be used in proofs. They are not forbidden like in Russell's Theory of Types. However we can't use non-stratified formulas to prove the existence of sets. In general, if I remember you prove that the sets exist by a different argument, instead of trying to appeal to stratified comprehension which would be hopeless if the formula is non-stratified.

If I am correct, in your example, you have $A \cup B = \{x : x \in y \vee y\in x\}$. You would instead write $A \cup B = \{x : x \in z \vee y\in x\}$ for $A=\{x : y\in x\}$ and $B=\{x: x\in z\}$, so that you can assign 1 to $y$, 2 to $x$ and 3 to $z$, this is stratified so now the union exists by stratified comprehension. But this does not exclude the case $z=y$ so we have the set $A \cup B = \{x : x \in y \vee y\in x\}$ even though $x \in y \vee y\in x$ is non-stratified.

A similar example is done in the book Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction (Springer Monographs in Mathematics), page 233.

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