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Regarding my comment that results were often first proved using transfinite induction, see my 25 November 2005 sci.math post in the thread titled "Transfinite exhaustion" and my 1 September 2006 sci.math post in the thread titled "No irrationals".

Also, you may want to search my posts in sci.math that mention Sierpinski (some, but not all, of these posts can be found using this Google archive search and this Math Forum archive search). For example, in this 3 January 2005 sci.math post in the thread titled "outer measure of the Vitali non-measurable set", I mentioned the following:

Sierpinski's 1965 book Cardinal and Ordinal Numbers also discusses various implications that hold between the statements $AC(*,n),$ where $n$ is a positive integer and $AC(*,n)$ (my notion) is the statement that the Axiom of Choice holds for an arbitrary collection of sets each having cardinality $n.$ See my 30 April 2007 sci.math post in the thread titled "I don't like the Axiom of Choice". Incidentally, the following comment (written by me), which appears at the beginning of that post, was about Gregory H. Moore's book Zermelo's Axiom of Choice: Its Origins, Development, and Influence, and thus I was happy to learn from this 1 March 2013 Math Stackexchange post that Moore's book is being reprinted by Dover. In my case, back in Fall 2008 I managed to obtain for about $100 (after an online search) a very good copy of the original 1982 hardcover edition of Moore's book.

Regarding my comment that results were often first proved using transfinite induction, see my 25 November 2005 sci.math post in the thread titled "Transfinite exhaustion" and my 1 September 2006 sci.math post in the thread titled "No irrationals".

Also, you may want to search my posts in sci.math that mention Sierpinski (some, but not all, of these posts can be found using this Google archive search and this Math Forum archive search). For example, in this 3 January 2005 sci.math post in the thread titled "outer measure of the Vitali non-measurable set", I mentioned the following:

Sierpinski's 1965 book Cardinal and Ordinal Numbers also discusses various implications that hold between the statements $AC(*,n),$ where $n$ is a positive integer and $AC(*,n)$ (my notion) is the statement that the Axiom of Choice holds for an arbitrary collection of sets each having cardinality $n.$ See my 30 April 2007 sci.math post in the thread titled "I don't like the Axiom of Choice". Incidentally, the following comment (written by me), which appears at the beginning of that post, was about Gregory H. Moore's book Zermelo's Axiom of Choice: Its Origins, Development, and Influence, and thus I was happy to learn from this 1 March 2013 Math Stackexchange post that Moore's book is being reprinted by Dover. In my case, back in Fall 2008 I managed to obtain for about $100 (after an online search) a very good copy of the original 1982 hardcover edition of Moore's book.

Regarding what Kuratowski says, I think his notion of a well defined base of a topology means that we are permitted to assume, in determining whether a proof is effective, that a fixed base has been provided to us in advance. However, we do not assume any of the present-day notions of recursive presentation of the base, such as one finds in Moschovakis' 1980 book Descriptive Set Theory (Chapter 3B:Recursive presentations, pp. 128-135; also on pp. 96-101 of the 2009 2nd edition).

Regarding what Kuratowski says, I think his notion of a well defined base of a topology means that we are permitted to assume, in determining whether a proof is effective, that a fixed base has been provided to us in advance. However, we do not assume any of the present-day notions of recursive presentation of the base, such as one finds in Moschovakis' 1980 book Descriptive Set Theory (Chapter 3B:Recursive presentations, pp. 128-135; also on pp. 96-101 of the 2009 2nd edition).

Note: The discussions about Sierpinski's and Luzin's notions of effectiveness (pp. 54-59) in the 1958 edition are written from a more classical point of view, a view that I believe better captures the spirit of the era in which Luzin and Sierpinski worked, than the corresponding discussion in the 1973 edition (pp. 67-73).

Petr Sergeevich Novikov, On effectively nondenumerable sets (Russian), Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 1939, 35-40. Zbl 24.30103; JFM 65.1169.04 [There is a French summary of Novikov's paper.]

Sierpinski's 1965 book Cardinal and Ordinal Numbers also discusses various implications that hold between the statements $AC(*,n),$ where $n$ is a positive integer and $AC(*,n)$ (my notion) is the statement that the Axiom of Choice holds for an arbitrary collection of sets each having cardinality $n.$ See my 30 April 2007 sci.math post in the thread titled "I don't like the Axiom of Choice". Incidentally, the following comment (written by me), which appears at the beginning of that post, was about Gregory H. Moore's book Zermelo's Axiom of Choice: Its Origins, Development, and Influence, and thus I was happy to learn from this 1 March 2013 Math Stackexchange post that Moore's book is being reprinted by Dover. In my case, back in Fall 2008 I managed to obtain for about $100 (after an online search) a very good copy of the original 1982 hardcover edition of Moore's book.

Whyburn, Review of Sierpinski's Lecons sur la Nombres Transfinis, Bulletin of the American Mathematical Society 36 #3 (March 1930), 175-176.

Regarding what Kuratowski says, I think his notion of a well defined base of a topology means that we are permitted to assume, in determining whether a proof is effective, that a fixed base has been provided to us in advance. However, we do not assume any of the present-day notions of recursive presentation of the base, such as one finds in Moschovakis' 1980 book Descriptive Set Theory (Chapter 3B:Recursive presentations, pp. 128-135; also on pp. 96-101 of the 2009 2nd edition).

Note: The discussions about Sierpinski's and Luzin's notions of effectiveness (pp. 54-59) in the 1958 edition are written from a more classical point of view, a view that I believe better captures the spirit of the era in which Luzin and Sierpinski worked, than the corresponding discussion in the 1973 edition (pp. 67-73).

Petr Sergeevich Novikov, On effectively nondenumerable sets (Russian), Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 1939, 35-40. Zbl 24.30103; JFM 65.1169.04 [There is a French summary of Novikov's paper.]

Sierpinski's 1965 book Cardinal and Ordinal Numbers also discusses various implications that hold between the statements $AC(*,n),$ where $n$ is a positive integer and $AC(*,n)$ (my notion) is the statement that the Axiom of Choice holds for an arbitrary collection of sets each having cardinality $n.$ See my 30 April 2007 sci.math post in the thread titled "I don't like the Axiom of Choice". Incidentally, the following comment (written by me), which appears at the beginning of that post, was about Gregory H. Moore's book Zermelo's Axiom of Choice: Its Origins, Development, and Influence, and thus I was happy to learn from this 1 March 2013 Math Stackexchange post that Moore's book is being reprinted by Dover. In my case, back in Fall 2008 I managed to obtain for about $100 (after an online search) a very good copy of the original 1982 hardcover edition of Moore's book.

Whyburn, Review of Sierpinski's Lecons sur la Nombres Transfinis, Bulletin of the American Mathematical Society 36 #3 (March 1930), 175-176.

Regarding what Kuratowski says, I think his notion of a well defined base of a topology means that we are permitted to assume, in determining whether a proof is effective, that a fixed base has been provided to us in advance. However, we do not assume any of the present-day notions of recursive presentation of the base, such as one finds in Moschovakis' 1980 book Descriptive Set Theory (Chapter 3B:Recursive presentations, pp. 128-135; also on pp. 96-101 of the 2009 2nd edition).