Sorry for the poor question wording. I have a vague memory of reading about a theory in philosphy, mathematics, or physics which says something like this. The idea, as I remember it, is that any given system (language? theoretical framework?) cannot be used to define a system larger than itself.

Godel's incompleteness theorem is maybe related, but not the same. This could be what I was thinking of, but does the above ring any other bells? If not, is a more precise version of the question even true?

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    Yes, this doesn't rise to the level of an answer, but I do suspect you are thinking either of Gödel or perhaps of one of the many mistaken presentations or mentions of Gödel's Incompleteness that suggest that it says something like what you're saying. Because after all, it's false. A few axioms like those of Euclidian geometry, can easily define an extraordinary range of shapes and figures for which it would be hard to say that they are “smaller” or the same size as those axioms. Jan 24 '15 at 4:47
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    Perhaps Hayek, the economist. He defended liberalism on the ground that human beings cannot grasp the functioning of their own society, because the system is bigger (hence trying to control the economy from a central state is hopeless). It's only a conjecture (he does not prove anything). Jan 24 '15 at 10:53
  • Another example along the line of ChristopherE's: Second order Peano arithmetic defines the natural numbers. Of course, a defender of your proposition could always quibble about what counts as "larger" and what "define" means, but that would only tell us that the proposition is vague to the point of meaninglessness in the first place.
    – WillO
    Jan 24 '15 at 17:42
  • Reminds me of Bhagavad Gita (9. 4-5.) "By Me, in my unmanifested form, are all things in this universe pervaded. All beings exist in Me, but I do not exist in them...And yet the beings do not dwell in Me--behold, that is my divine mystery. My spirit which is the support of all beings and the source of all things, does not dwell in them." Jan 25 '15 at 15:03

On an anecdotical level, maybe you think of the quote about the human brain by Emerson W. Pugh:

If the human brain were so simple that we could understand it, we would be so simple that we couldn't.

I.e. the human brain is a system incapable of describing something as complex as itself, let alone something larger.


I suspect you are thinking of Russell's Paradox (1901).

Let S be the set of all sets which are not members of themselves.

Is S a member of itself?

If not then by the definition of S, S is a member of S. But if S is a member of S then by definition it is not a member of itself!

So, we have a contradiction and therefore there must be something illegitimate about defining sets of sets which are or are not members of themselves.


You may be thinking of the 'downward' one of the Lowenheim-Skolem theorems. An axiomatization cannot require a model that is strictly larger than the set of sentences generated by the language. Most importantly, every countable axiom set has a model with only countably many elements.

So in particular, any countable axiomatization of the Real numbers has a model with countably many elements. But the real numbers as we use them are both uncountable and well-defined, they are just not the smallest equivalent model to whatever view we take of the Reals at any given point in time.

But unless you drag infinite cardinals into it, this seems directly contradicted by the notion of language. Finite human languages certainly define a world that includes a lot more than the language, if only to a limited degree. Theoretically, every language is countably infinite in its number of sentences. But the actual sentences that can be meaningfully understood, given the limitations of human intelligence are a finite subset of the physical things. The number that will ever be said is even smaller.

And this is true of specific mathematical languages. Clearly the finite axiom set of ZFC, which is finite, defines an infinite world of sets. The finite algebraic model of the integers defines an infinite sequence. And so on.

So as a principle, this may be true in some situations. But even then, it is demonstrably not true of some finite things.

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