GPT-2, GPT-3, Codex, Fine-tuning, Davinci, Curie

Is davinci being not available for fine-tuning right now because of the size of the base model(GPT-3, 175B)?
Extrapolating from this, are curie and other cheaper engines based on GPT-2 or earlier models? or GPT-3 but smaller datasets and a smaller number of parameters?
What are Codex’s base model and engine? or is Codex a completely different model?

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We’ll soon support davinci fine tuning. No, the other models are gpt3 based. Codex is trained on code.

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Thanks for clarifying the distinctions, @boris.