# Axiomatic system or Set theory [duplicate]

Which one is more fundamental, Set theory or Axiomatic system? Which one can be defined without the other?

## marked as duplicate by TravisJ, s.harp, Asaf Karagila♦ set-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jan 24 '17 at 19:42

• No, it does not. Predicate counting are how exists and forall behave without sets let us say $(\forall x) \Longrightarrow (\exists x)$ and this x has some property P, which does not need to be enough to form a set, you can search that in elementar logic, there are counterexamples that not every property is coming to form a set. – nikola Jan 24 '17 at 21:01
• Forall and exists address to, well, some property P of given element $x$, if you tell that something holds for all $x$, then you have said that all of those $x$ have that property P. Now, let us have the sentence $(\forall x \in X) x \in x$, that is totally correct if you are relying on the predicate account exactly, but if you form a set with that property, you will end up in contradiction. – nikola Jan 26 '17 at 15:29