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Berhard Bolzano (1781-1848) made great advancements in the theory of sets. Bolzano expanded on Galileo's findings and provided more examples of this theme. One of the most respected mathematicians of all time is Karl Friedrich Gauss. Gauss gave this insight on infinity:
As to your proof, I must protest most vehemently against your use of the infinite as something consummated, as this is never permitted in mathematics...
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Gauss gave this insight on infinity:
As to your proof, I must protest most vehemently against your use of the infinite as something consummated, as this is never permitted in mathematics. The infinite is but a figure of speech; an abridged form for the statement that limits exists which certain ratios may approach as closely as we desire, while other magnitudes may be permitted to grow beyond all bounds...
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