Bertrand Russell

It is hard to imagine anyone who contributed more, or detracted more, from recent  Intellectual History than the indefatigable Bertrand Russell (BR).  

Co-authored with Alfred North Whitehead, Russell’s Principia Mathematica brought Issac  Newton into the 20th century. Plus, BR was the first major thinker to throw a flag on ‘the  calculus’. Discovered simultaneously by Newton and Leibniz c. 1700, calculus was  assumed to provide the ‘final answer’ to Zeno’s pesky paradoxes. It doesn’t!  

Russell pointed out that calculus is a computational method useful for making  calculations about real phenomena but not for modeling those phenomena per se. The  idea that ‘calculus’ describes something real about the world is an example of what  colleague Whitehead would have called, the fallacy of misplaced concreteness. Bottom  line: Zeno lives on…and we have Russell to thank for pointing that out. 

Finally, BR’s tireless activism in pursuit of World Peace, while often misdirected, cannot go  unmentioned. 

On the other hand, Russell resurrected the ‘Problem of Evil’ as a rationale for his atheism  (Why I am not a Christian) and his reformulation of set theory, while insightful, set back progress in this field by half a century. It is these last two aspects of his legacy that concern  us today and we will attempt to show that, surprisingly, the two are closely related. 

Russell noticed that self-referential propositions can be inherently self-contradictory.  Consider, for example, the proposition: ‘All propositions are false’. There is an inherent  contradiction here. If ‘all propositions are false’, then the proposition, ‘all propositions are  false’ must itself be false; and if ‘all propositions are false’ is false then some propositions  must be true, which violates the premise; ergo contradiction. 

So, Russell MacGyvers the problem! He adds an axiom now known as the Axiom of  Foundation to Set Theory; it states that sets cannot be elements of themselves. So ‘all  propositions are false’ is not in fact a well-formed proposition because it designates a set,  the set of all propositions, that contains itself as an element.  

Now if a set cannot be an element of itself, then what appears to be a self-referential  proposition is not even a proper proposition. Inconveniently however, the real World does not step aside, even for Bertrand Russell. 

Consider, for example, the set of all mathematical objects; such a set would itself appear  to be a mathematical object in which case it would be an element of itself and no  contradiction results, in bold faced defiance of BR’s dictum. Some nerve! When the ideas  of an intellectual clash with reality, reality is supposed to step aside. Cheeky sets!  

But Russell is unphased. If ‘Set A’ contains itself as an element, then ‘Set A’ cannot be a set  after all. It may walk like a set and it talks like a set, but it cannot be a set. ‘Why? Because BR says so!’ 

In his best imitation of a 1950’s parent, Russell stifles all dissent. Surprisingly, we, like  cowed children fearing a smack, sheepishly comply. But if Set A is not a set, what is it? It’s  a ‘class’, obviously…do try to keep up! (If you have the feeling that you’re caught in an ‘Alice  story’, you’re not wrong.) 

In Genesis, neither God nor man (Adam) has the power to create something just by  speaking its name. That’s why Principia Mathematica had to replace Torah. Just as a simple  ‘abracadabra’ in the mouth of a certified magician can turn lead into gold, so Russell, a  certified logician, was able to turn absurdity into truth. Amazing how far we’ve come in 3  millennia! 

But ‘you cannot fool all of the people all of the time’ – A. Lincoln. From the get-go a few  brave souls, living like Obi Whan Kabobi on the edge of civilization, dared to raise doubts. “I  don’t see any clothes here, do you?” But it was not until the 1980s that Peter Aczel (PA) led a no holds barred assault on Russell…and vanquished him. 

Thanks to PA, we now understand that the Axiom of Foundation is not a necessary part of  Set Theory, that we can get along just as well without it and open a huge new universe of  possibility in the process. Aczel reformulated Set Theory to include self-referential sets  without generating any contradiction. Sets that seemingly obey the Axiom of Foundation  form a tiny subset of all sets.  

Analogy: a 3 year old is understandably proud; she ‘knows her numbers’, i.e. the natural  numbers from 1 to some upper limit. Imagine her surprise when she discovers that Natural  Numbers form a tiny subset of Real Numbers which are a tiny subset of Hyperreal  Numbers, etc. 

While we will not get into this today, I would argue that a Universe without self-reference  could not exist at all or, if it did, it would be a lifeless wasteland (because what is ‘life’ but  self-reference?). Genesis: “…the earth was without form or shape, with darkness over the  abyss and a mighty wind sweeping over the waters.” That’s us, according to the Axiom of  Foundation.

In 1908 Russell stated the rule that held logic in thrall for nearly a century: “Whatever  involves all of a collection must not be one of the collection.” And so, of course, Russell  could not possibly be a Christian…and the Problem of Evil has nothing to do with it. Sorry,  Bertrand! Next time around, “Know thyself!” 

Turns out, the huge intellectual edifice known as ‘Christianity’ has its own Fundamental  Axiom, but it’s the opposite of Russell’s: “There is at least one collection that is one of the  collection.” In other words, there is at least one set that is a proper member of itself.  

I call that the Axiom of Incarnation. This is the non-negotiable foundation of Christianity:  God, Being per se (Exodus 3: 14), who contains everything that is, is also a quantum of  being (Jesus, the Christ) among what is. God contains God! Being at its most fundamental  level is self-referential. And so a Universe is possible after all. 

Cosmologists claim to be able to account for the emergence of a Universe without  resorting to the God Hypothesis; maybe so! But there can be no Universe without the Axiom  of Incarnation. 

Self-reflective sets are also recursive. The set acts on itself. Examples of recursive process  abound in the stories and doctrines of Christianity. To cite just a few: 

➢ “Love your neighbor as yourself.” (Mt. 22: 40) 

➢ “Whoever eats my flesh and drinks my blood remains in me and I in them.” (Jn: 6: 56) ➢ “Blessed are the merciful for they will obtain mercy.” (Mt. 5: 7) 

➢ “Forgive us our trespasses as we forgive those who trespass against us.” (Mt. 6: 12) ➢ “I am in my father, and you are in me, and I am in you.” (John 14: 20) 

My own sense is that the Axiom of Incarnation does not go far enough. I propose replacing  the formulation above with something stronger: “A set is a collection that is one of the  collection.” In other words, every proper set is a member of itself. I call this the Axiom of  Trinity and I will explore it further in future posts. 

But we have already covered a lot of ground. Starting with Russell’s claim that no set can be  a member of itself, we proposed a set, called Incarnation, that falsifies Russell’s claim.  Freed from the shackles of Foundation, can we proceed from Incarnation to Trinity? (BTW,  the early Church, up to the Council of Nicaea (c. 325 CE), struggled with this same  question.) Assuming Incarnation, can we prove Trinity using Set Theory? I don’t know yet.  Stay tuned!

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M.C. Escher’s Drawing Hands (1948) depicts two hands sketching each other into existence, visually capturing the self-referential loop at the heart of logical paradox and the Incarnation.¹

¹Scanned from The Magic of M. C. Escher Artist: M. C. Escher Year: 1948 Medium: Lithograph Dimensions: 28.2 cm × 33.2 cm (11.1 in × 13.1 in) Preceded by: Up and Down (1947) Followed by: Dewdrop (1948)