A Note on some New Infinity Puzzles

Abstract

In this short note I argue that, using the type of configurations put forward in a recent paper by Laraudogoitia in this same journal, new paradoxes of infinity of a completely different nature can be formulated.

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Notes

  1. 1.

    This is eminently intuitive: at higher pressure volume decreases and at lower pressure volume increases.

  2. 2.

    Important physical theories applicable to the real world does not mean true physical theories, rather theories that provide a close approximation to the real behavior of the world. The abundant literature now available on supertasks makes use of precisely these kinds of theories.

  3. 3.

    “an ideal gas is still the best “laboratory” for understanding materials with non-trivial thermodynamics” (Lautrup 2005, p. 52). This justifies the philosophical relevance of considering possible worlds where Boyle’s law (a characteristic law of ideal gases) is fulfilled, in the same way that it is philosophically relevant to consider possible worlds where Newton’s laws (characteristics of classical mechanics) are fulfilled. Seen in a complementary way, a gas that strictly obeys Boyle’s law is a scientific idealization, but it is as theoretically important as the notion of a perfectly rigid body. And just as there is a whole chapter on thermodynamics (ideal gas thermodynamics) which deals with gases that fulfill Boyle’s law, there is also a branch of venerable tradition in classical mechanics which deals with rigid bodies.

  4. 4.

    The name “serrated continuum” is introduced in Benardete 1964. It refers in the present case to the division of a finite material body into a denumerable infinity of finite parts but which decrease in size. And the interesting aspect results from the realization that this infinity of finite parts has a natural order with no final element.

  5. 5.

    In this infinite sum, subscript i ranges from - ∞ to + ∞, with the exception of i = 0. Thus, ∑V°(Ci) = ∑(1/i2) is the sum of two ordinary infinite series, namely, ∑V°(C-i) and ∑V°(Ci), and i ranges from +1 to + ∞ in each of them. The same considerations apply to all the infinite sums in sections 1 and 2 of the paper.

  6. 6.

    Benardete 1964 describes it in these words: “Here is a book lying on a table. Open it. Look at the first page. Measure its thickness. It is very thick indeed for a single sheet of paper - 1/2 in. thick. Now turn to the second page of the book. How thick is this second sheet of paper? 1/4 in. thick. And the third page of the book, how thick is this third sheet of paper? 1/8 in. thick, &c. ad infinitum. ... . Close the book. Turn it over so that the front cover of the book is now lying face down upon the table. Now - slowly - lift the back cover of the book with the aim of exposing to view the stack of pages lying beneath it. There is nothing to see. For there is no last page in the book to meet our gaze.” (pp. 236–237) “... But how can we describe this horror, this ineffable entry of ours into the open end of the Z-series?” (p. 238)

  7. 7.

    If there was one single Z-series of walls (and compartments) then it would be easy to see that final equilibrium would not be possible. Let us suppose that initially C contains only compartments C1, C2, C3, ... Cn, ... (which fill its entire interior). In the assumed final state of equilibrium (with null volumes and infinite pressures, as we know) the gas in C1 would push C to the left with infinite force, but no force would be exerted on C to the right because all the gases would be bunched together to the left of C! This pathology is avoided with the two Z-series of walls, given the symmetry of the situation.

  8. 8.

    Compare this to what Benardete 1964 says about paradoxes such as the paradox of the book: “... it is not the infinitude of the infinite open-ended Z-series that alarms us but rather its uncanny open-endedness.” (p. 256)

  9. 9.

    What I call here paradoxes of the serrated continuum roughly correspond to what Rayo (2019) calls “omega-sequence paradoxes” and “reverse omega-sequence paradoxes”

References

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Acknowledgments

Research for this work is part of the research project FFI2015-69792-R (funded by the Spanish Ministry of Economy and Competitiveness, Government of Spain).

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Correspondence to Jon Pérez Laraudogoitia.

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Laraudogoitia, J.P. A Note on some New Infinity Puzzles. Philosophia 48, 1483–1491 (2020). https://doi.org/10.1007/s11406-019-00141-0

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Keywords

  • Infinity paradoxes
  • Equilibrium
  • Boyle’s law