on russells paradox

russell’s paradox is paradoxical – but not in that way

overview

Russell’s paradox is held as an example of why set-theoretic sets cannot not contain themselves – the paradox depends upon a set, which is the result of a specifically crafted predicate, containing itself, in violation of respective predicate constraints.

#todo plainly describe Russell’s paradox

However fun and engaging russell’s paradox is – by the time the specifically crafted predicate enters the scene (of russell’s example), a more fundamental paradox (of a set containing itself) has already occurred.

This fundamental paradox can be described in simpler terms, by making explicit some of the hidden assumptions (implicit within set-theory), which illustrate a peculiar inconsistency in the fabric of set-theory itself.

key terms: time; causality; dependency; declaration; definition; realisation; mutability, consistency

the premise

Consider a set-theoretic set as state, and the result of a process:

1. Whereby a set-theoretic predicate filters some prior-state (also a set)
2. Such that a new artefact is created, which contains {all; some, or; none} of the contents of the prior-state

a set-theoretic set which is the result of a predicate, implicates prior state – a preceding set – which at root, is defined by each set-theoretic forms’ rules and axioms

Necessarily:

1. Prior-state must exist (and be well-formed) before any formal operational process, like predicate evaluation, can occur
   • Because if the process occurs before prior-state is:
     • Available, what is processed?
     • Well-formed, is processing consistent?
   • Predicate evaluation is dependant upon well-formed prior-state
2. Resultant-state cannot exist before the process (of predicate evaluation) has occurred
   • Resultant-state is dependent upon respective process of predicate evaluation
     • Which is dependent upon process initiation
       • Which is dependent upon available, well-formed prior-state
   • This sequence describes temporal and operational constraints, inherent in set-theoretic consistency
3. Resultant-state cannot contain anything which was not contained within pre-existing, well-formed, prior-state
   • Is it not fundamentally inconsistent for set content to ‘spontaneously appear’?

Summary:

1. A set cannot contain itself, because it did not exist before the process which created it occurred
2. Specifically, a set did not exist in time to be included in the pre-existing well-formed prior-state, which the process which created it (the set) depended upon, as a pre-requisite for operation

“russells paradox is itself built upon a more fundamental (temporal and operational) paradox, and as such, seems invalid, and redundant – is this correct?”

paradoxes are contrived

For a set to contain itself, the fundamental mechanics of set-theory must either:

1. Allow for resultant sets to be populated by secondary means ¹, or
2. Allow for a type of forward-declaration, whereby some placeholder for a yet-to-be created resultant-state is injected into prior-state in advance of presentation to the operational process which depends upon the prior-state to generate the resultant-state; and subsequently, upon process completion (but before general availability of the resultant-state), access and mutate the placeholder, such that the placeholder then identically mirrors the resultant-state, which is a set
   • Which seems abhorrent for many reasons, including:
     1. Implicit, undeclared behaviour of placeholder sets being injected into sources of truth to facilitate downstream needs
     2. Set mutability, by undeclared, non-explicit operational side-effect

—though perhaps the placeholder is simply an empty set, later populated?

Even so, the operational mechanics of the process of predicate evaluation (which manifests consistency throughout set-theory), still cannot access the placeholders future nature, to determine and evaluate the predicate condition.

And quite curiously, this take is also consistent with the outcome of Russell’s paradox – when conditions are indeterminable, and outcomes forced, ought we not expect undefined or undefinable behaviour? And what is undefined or undefinable behaviour in place, if not a formal paradox of consistency?

We might observe that one paradox, wishfully or blindly worked around, simply leads to another; and with it, observe a formal genesis of an eventual, web-of-lies.

conclusion

A set cannot contain itself:

• Because it did not exist before the process which created it occurred
• Because fundamental causal constraints of our reality, which are implicit within very the fabric and fundamental operations of set-theory, simply have have no way to do so without introducing downstream inconsistencies ²

A set cannot contain itself, because a set containing itself, is itself a paradox – fundamentally (temporally, and operationally) inconsistent.

Russells paradox is perhaps not the paradox that you thought it was.