First and Second modes of generations and limitation

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The discussion centers on Joseph Dauben's exploration of George Cantor's concepts of the first and second modes of generation, along with the principle of limitation. Participants are encouraged to share their interpretations and descriptions of these concepts to enhance understanding. The principle of limitation is highlighted as a primary focus, with a request for varied explanations. The aim is to clarify these mathematical philosophies in more relatable terms. Engaging with these ideas can deepen comprehension of Cantor's contributions to mathematics and infinity.
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In Joseph Dauben's book on George Cantor, his philosophy of the infinite and hsi mathematics...there is a section which talks of the first mode of generation, the second mode of generation and the principle of limitation.


Any words, any of you would like to use to describe all three of these I would be appreciative of. I just wish to hear them explained in different ways and in your own words, so it will catch on more. Mainly, the principle of limitation is what I'm most concerned with

Thanks
 
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Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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