If we cannot prove consistency from within, the conditional 'if consistent, then the Gödel sentence is true' provides no categorical grounds for asserting the sentence's truth—only its truth relative to an assumed but unverifiable hypothesis.

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Categorical grounds(as opposed to conditional or relative grounds): Solid, definite proof or reasons that work on their own, without needing to assume anything else first.
Conditional statement(the 'if consistent, then...' structure in the quote): A 'if-then' claim: 'if X is true, then Y must be true' — it doesn't say X actually IS true, just what follows IF it were.
Consistency (in logic)(as the property being discussed in the conditional statement): When a set of rules or statements don't contradict each other — you can't prove both a statement and its opposite at the same time.
Gödel(as a historical figure in mathematical logic): Kurt Gödel was a 20th-century mathematician and logician who proved that any consistent formal system (a set of logical rules) is incomplete—meaning there are true statements it can't prove.

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