So I have been pondering about language. By language L I just mean a series of symbols. The upper limit of this series of symbols is Aleph-zero. Yet somehow using these symbols the human is able to understand the meaning of phenomena of cardinality Aleph-one. Let us denote the meaning of phenomena as M. How is this happening (what is the possible mappings)? The cardinality of L is Aleph-zero, that of the phenomena Aleph-one and M(Aleph-one) must be Aleph-zero?

So this seems to have interesting possibilities:

  • M is not a well defined operation and cannot be expressed in mathematics (rather should be taken for granted)
  • There is no one to one mapping between L and M(Aleph-one). We humans do not understand meaning (solely) through language
  • There is no one to one mapping between M(Aleph-one) and Aleph-one. Certain meanings are redundant.
  • There can be truths about the phenomena of Aleph-one which cannot be captured in L.

Here's an example:

I have a friend Bob who speaks only in morse code. After a long time I explain him to him what rational numbers are. He understands he concept. A natural question comes does my friend understand this concept in it's entirety. Perhaps there are things he doesn't get. But I can talk more morse and he (perhaps) will get it. How do a countable set of symbols make Bob understand phenomena which is uncountable.

  • Mathematics cannot ever formally describe meaning*
  • Well maybe Bob used innate knowledge and was inspired by speech to bring it forth.
  • While the phenomena of rational numbers can be of cardinality aleph-one the meaning behind the rational numbers reduces to phenomena of cardinality to aleph-zero.
  • There are truths about the rational numbers that no amount of symbols will ever make sense on why it is so.


Are there any possibilities I have missed? What answer(s) are correct?

*Please do not ask me to formally describe meaning since I kind of subscribe to this line of attack.

  • What has Morse Code to do with it? It's just a painfully slow way to speak in [some human language], but a reliable means over very long distances. Commented Aug 10, 2023 at 18:56
  • I thought it made the language as a string of symbols of cardinality aleph zero more obvious (was trying to give a good example) Commented Aug 10, 2023 at 18:58
  • so your question is how can we describe higher cardinalities using a language that has at most aleph null wffs?
    – emesupap
    Commented Aug 10, 2023 at 19:16
  • 2
    See Kripkenstein on rules for a fleshed out version of this argument to meaning more generally. Also Quine on meaning skeptism. Then you can read the standard responses. In response to the questions about cardinalities specifically, see perhaps Skolems paradox. Also see second order logic, which features many models up to isomorphism. plato.stanford.edu/entries/logic-higher-order/…
    – emesupap
    Commented Aug 10, 2023 at 19:30
  • 1
    Are you sure the cardinality of rational numbers is ℵ1? As for meaning if you're a platonist or structuralist, you can always ground math statements to certain categorical set of mathy objects or structures in a 1-1 or 1-many fashion (meaning interpretations may not be unique) and in a judgmental or normalization manner. Even for a formalist, at least above 'grounding' process can be interpreted as state affairs of a game. If you're more ambitious to include all other meanings you have to premise on more ontological philosophies where not everything can be formally described by math alone... Commented Aug 11, 2023 at 23:56

2 Answers 2


Your question is reminiscent of Skolem's paradox:

Skolem's Paradox involves a seeming conflict between two theorems from classical logic. The Löwenheim-Skolem theorem says that if a first-order theory has infinite models, then it has models whose domains are only countable. Cantor's theorem says that some sets are uncountable. Skolem's Paradox arises when we notice that the basic principles of Cantorian set theory—i.e., the very principles used to prove Cantor's theorem on the existence of uncountable sets—can themselves be formulated as a collection of first-order sentences. How can the very principles which prove the existence of uncountable sets be satisfied by a model which is itself only countable? How can a countable model satisfy the first-order sentence which says that there are uncountably many mathematical objects—e.g., uncountably many real numbers?

A full answer to your question would require perspicuous definitions of understanding and word meaning that for now, perhaps, much escape us. We do have the ability to think over infinitary logics along with other matters of higher set theory, to think through sequences of greater and greater cardinality. Arguably, plural quantification even does much of the needed work: we can have the expression "∃rr" in play, indeterminately ranging over the (indeterminate) manifold of the alephs (say).

But then again, what is it for a symbol to mean something, and how many symbols are allowable? In principle, we could use a continuous stream of percepts symbolically, could have continuously many symbols, but repeating our use of percepts in this way would not comprise a very stable language.

So there's an aspect of the problem: not mere meaningfulness "just like that" but the stability thereof, the communicative recurrence of the icons we intend to use to convey information to ourselves and each other. At any given time, we will have ever written down only finitely many notations of individual alephs, even as we skip up the cumulative hierarchy to talk about having k-many of this or that large cardinal. Our ability to target absolute infinity in our sights, and the appearance of openness that this "thing" has to its name (it is open under all other predicates besides "is open under all other predicates"), with absolute infinity being yet seemingly "one thing," gives us the capacity to think over all the alephs at once, we wish we could unhesitatingly contend. However:

At the core of the problem lies the assumption that the set-theoretic paradoxes cast doubt upon the existence of a comprehensive domain of all objects. What they reveal, according to Dummett (1991, 1993), is the existence of indefinitely extensible concepts like set, ordinal, and object. For Dummett, the indefinite extensibility of set is incompatible with the existence of a comprehensive domain of all sets, since no matter what putative domain of all sets we isolate, we find that we can employ Russell’s reasoning to characterize further sets that lie beyond the putative domain of all sets with which we began. The set of all non-self-membered sets in the initial domain cannot, on pain of contradiction, be in that domain, which means that it must lie in a more comprehensive domain of all sets. If there is no domain of all sets, there is, the thought continues, no hope for a domain of all objects. One may respond to this line of argument by contesting Dummett’s diagnosis of the set-theoretic antinomies. Boolos (1993) suggests that we take Russell’s paradox, for example, to establish that not every condition determines a set. The moral of Russell’s paradox is that there is no set of all non-self-membered sets, not that it lies beyond the initial domain of quantification.

The Boolos just mentioned is the progenitor of plural quantification theory by the by, incidentally. For more on issues of indefinite/indeterminate extendibility, see Storer[10], esp. §§3.2, 3.3, 5, and 6.5.


Your puzzlement about the cardinalities of language and meaning is understandable, but rests on some false assumptions. You wrongly equate the external forms of language with the underlying grammatical competence that enables our infinite expressive capacity.

Our recursive merge operation allows us to build linguistic structures of unbounded complexity from finite elements. So our innate universal grammar is capable of producing an infinite set of meaningful utterances, not limited by countable vocabularies.

Meaning is not a mathematical function - it does not stand in neat one-to-one correspondence with linguistic forms. It emerges through the interaction of language with other cognitive interfaces and contextual factors. Human thought itself outstrips what can be effably put into words.

Your friends' comprehension of rational numbers from morse code symbols does not imply language perfectly expresses meanings with matched cardinalities. It draws on their pre-existing mathematical concepts linked to the trigger of linguistic cues.

So no, truths about phenomena cannot be fully captured by mapping language onto them. Meaning is cognitive and conceptual, not just communicative. Grammar allows indefinite generation of meanings, but many remain ineffable. Math may gesture at realities language cannot encapsulate.

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