Does ZFC get rid of Russell’s paradox?

Many people today appear to believe that ZFC (Zermelo–Fraenkel set theory) gets rid of Russell’s paradox. They appear to believe that ZFC in some way solves Russell’s paradox.

This is a total misunderstanding of ZFC. What ZFC does is that it avoids Russell’s paradox in order to find a single consistent mathematics (ie, to connect discrete mathematics to traditional mathematics consistently). This aim is, apart from being an interesting mission in itself, of greatest importance for computer science, because computers are fundamentally discrete. Computer programs require such a mathematics to avoid Russell’s paradox (in practice infinite loops).

No, ZFC just hides Russell’s paradox in one place it can be hidden in a consistent mathematics. ZFC (there are several other possibilities) hides it in the fact that a single element of a set (also called an object) is not a set, but the product of the empty set. This hiding does not mean that ZFC has got rid of Russell’s paradox, but just that it produces a consistent mathematics by hiding it there. ZFC in itself is nothing but a consistent orthogonal system, like the Linnean systematics, although the Linnean systematics instead produces a consistent conceptualization.

Russell’s paradox has a deeper importance than the systems we can produce in meaning that we can choose what we want to be consistent with what among objects, finite classes and infinite classes, but that we can’t be consistent with all of them in relation to what they refer to (ie, what we talk about), because the paradox states that this possibility is fundamentally paradoxically contradictory.

The problem (ie, Russell’s paradox) with ZFC emerges when it is applied on reality, in that it lacks a consistent connection to reality. A single element of reality, ie, an object, is a product of the empty set in ZFC, whereas sets in ZFC (consistent in this system) are paradoxically contradictory in reality. It means that although ZFC is consistent, it can’t connect to reality consistently (ie, can’t be existentially consistent). It is thus good for computers, but not for discussing reality, ie, can model reality ambiguously, but can’t find a single truth.

The misunderstanding that ZFC (Zermelo–Fraenkel set theory) gets rid of Russell’s paradox may lead to the conclusion that a single truth, like “the tree of life” and “Higgs particle”, is possible, but this is thus a misunderstanding. ZFC does not open the door to a single truth, but just produces a single consistent mathematics, nothing more and nothing less. ”The tree of life” and “Higgs particle” are just as impossible without ZFC as they are with ZFC. ZFC makes no difference to the fact that they are impossible..