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. Aug 10 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) Aug 10 at 18:58
  • so your question is how can we describe higher cardinalities using a language that has at most aleph null wffs?
    – emesupap
    Aug 10 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
    Aug 10 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... Aug 11 at 23:56

1 Answer 1


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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