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So, from what little I have read (such as this answer), it appears to be that one reason why the program of Logicism, as laid out in the Principia Mathematica, failed was that its goals (of finding a consistent and complete foundation of mathematics by showing it to be derivable from logic) as a result of them being shown to be impossible as a result of Gödel's incompleteness theorems.

However, I still do not understand why this does, to such an extent, defeat Logicism, whilst still allowing for the use of, for example, ZFC. Whilst I do have a vague understanding of this being as a result of how the Principia seems to lack the distinction between provability and truth, as apparent from this quote from a paper from the bulletin of symbolic logic:

"Whitehead and Russell fail to distinguish between the concept of truth and that of provability."

I still cannot precisely understand what it is, in this lack of distinction, that makes Logicism so vulnerable to the incompleteness theorems.

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ZFC is susceptible to Gödel's incompleteness as was the Principia Mathematica.

See Gödel’s Incompleteness Theorems for an introduction to the theorem :

Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.

Assuming the consistency of ZFC, Gödel’s Theorem applies to it.

Logicism was the project, started by Gottlob Frege, of founding arithmetic and analysis (not geometry) on "pure logical" principles.

W&R into their masterpiece Principia Mathematica completed the project but failed to use only "pure logical" principels, because they were forced to use the Axiom of Infinity and the Multiplicative axiom (aka : Axiom of Choice) whose mathematical grounding was indisputable (and they are used by ZFC) but whose "logical" nature was not teneable.

Assuming that the statement that "Whitehead and Russell fail to distinguish between the concept of truth and that of provability" is correct (more or less), this was not the source of the "failure" of Logicism.

In any case, "failure" must be taken with a pince of salt... there is nothing logically (and mathematically) wrong into the Principia.

You can see also the related post : What did Whitehead and Russell's “Principia Mathematica” achieve?

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