>>14022953>>14022984>>14023005(cont.)
One other thing. There are other "paradoxical" classes typically avoided in axiomatic set theory besides the Russell Class, including the Burali-Forti Class (class of all ordinals), and the Universal Class (class of all classes). These are often regarded as interchangeably, equally paradoxical with the Russell Class, but it's important to be aware that these other paradoxical classes do not exhibit the same property of being by themselves syntactically disprovable when straight-forwardly translated into FOL the way the Russell Class is.
Consider the Universal Class. If we translate its definition into FOL:
>?x?yMxyunless we package this together with the appropriate theory (e.g., set of sentences or axioms, like the FOL fomulations of the axioms of ZF/ZF+C), we can't disprove it.
Its negation:
>!?x?yMxyis *not* a theorem of first-order logic, meaning there are at least some models that satisfy:
>?x?yMxyFor example, consider a model of natural numbers 0, 1, 2..., and let "M" stand for "<=". Clearly, 0 is <= every natural number, including itself.
The Russell Class is paradoxical in that it contradicts the laws of classical logic itself, whereas other paradoxical classes only contradict particular axioms of particular axiomatic set theories, and there are in fact non-standard axiomatic set theories (like NF and NFU) that admit them as sets.
