# Qualitative versus quantitative representation: a non-standard analysis of the sorites paradox

## Abstract

This paper presents an analysis of the sorites paradox for collective nouns and gradable adjectives within the framework of classical logic. The paradox is explained by distinguishing between qualitative and quantitative representations. This distinction is formally represented by the use of a different mathematical model for each type of representation. Quantitative representations induce Archimedean models, but qualitative representations induce non-Archimedean models. By using a non-standard model of $${\mathbb{R}}$$ called $$the\,hyperreal\,numbers$$, which contains infinite and infinitesimal numbers, the two paradoxes are shown to have distinct structures. The sorites paradox for collective nouns arises from the use of infinite numbers, whereas the sorites paradox for gradable adjectives arises from the use of infinitesimal numbers. Each paradox can be traced to a different source of vagueness. The sorites paradox for collective nouns is caused by $$semantic \,\,indeterminacy$$, and the sorites paradox for gradable adjectives is caused by $$epistemic$$ $$indiscriminability$$. If correct, this analysis implies that infinite and infinitesimal numbers are cognitively real, and that they play a role in the semantic interpretation of natural language.

This is a preview of subscription content, access via your institution.

1. 1.

See Williamson (1994), and the discussion in Sect. 5 below.

2. 2.

The use of the term “the” hyperreal numbers is a bit sloppy, since unlike the real numbers there is no unique model up to isomorphism. However, for every cardinality there exists a unique model up to isomorphism. See Kanovei and Shelah (2004) for a more thorough discussion of the uniqueness of the hyperreal numbers. As an anonymous reviewer points out, moreover, the construction of the hyperreals requires the Axiom of Choice, and for our purposes weaker versions of a non-Archimedean structure may suffice.

3. 3.

Compare with Williamson (1994), who views the inductive premise as false.

4. 4.

The choice of the hyperreals here is arbitrary. In fact, any non-standard model of an ordered field (the ‘hyperrational numbers’ for example) would work. However, the question whether our mental representation of numbers is also continuous or just dense is orthogonal to the subject matter of this paper and will not be explored.

5. 5.

The compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model.

6. 6.

$$0 \in I$$ is necessary in order for the equivalence relation ‘$$\sim$$’ to be reflexive. This will be discussed further in Sect. 2.3.1.

7. 7.

Proofs of the properties listed in (9a-d) are provided in the Appendix. (9e) trivially follows from the fact that $$H - H = 0$$.

8. 8.

(9e) and (8a) are two properties that differentiate between the mathematical structures of infinitesimal and infinite numbers.

9. 9.

It is possible to use $${a - b}$$ instead of $$\left| {a - b} \right|$$, because of the definition of $$I$$ given in (7b).

10. 10.

I will rely on the following property of equivalence classes:

 (i) $$\left[ a \right] = \left[ b \right] \,\, {\text{iff}} \,a\sim b\, {\text{iff}}\, \left[ a \right] \cap \left[ b \right] \ne \emptyset$$
11. 11.

Proof of the lemma is provided in the Appendix.

12. 12.

Usually the notation used is $$a + I$$ instead of $$\left[ a \right]$$, however it is trivial to see that $$I = \left[ 0 \right]$$.

13. 13.

The proof that $${\mathcal{F}}/I \cong {\mathbb{R}},$$ and the fact that this is given by (19), is provided in the Appendix.

14. 14.

(15) ensures that $$g$$ is well defined.

15. 15.

This paper focuses on collective predicates that are count terms. Link’s framework is also applicable to mass terms, however. A reviewer asks whether the framework could be applied to sorites arguments involving mass nouns. Chierchia (2010), for example, considers that a spoonful of rice is rice, but asks whether a single grain of rice is rice. One may respond that one grain of rice is indeed rice (unlike one grain of sand relative to a heap of sand). Granting that the limit is unclear, however, the present framework is in principle applicable to such cases.

16. 16.

$$\langle D_{e} ,\sqcup ,{\sqsubseteq} \rangle$$ is isomorphic to $$\langle {{\mathcal{P}}}\left( {\left( {Atoms\left( {D_{e} } \right)} \right)} \right), \cup , \subseteq \rangle$$, where $${{\mathcal{P}}}$$ is the power set function, and is given by the mapping $$a \,\mapsto\, \left\{ a \right\}$$.

17. 17.

This can be formalized as (i):

1. (i)

$$forest \left( x \right) \to {\forall a \in Atoms\left( x \right)\left( {tree\left( a \right)} \right)}$$

18. 18.

This can be formalized as (ii), where $$P$$ is the set of two-dimensional planes:

1. (ii)

$$heap\left( x \right) \to \forall y \in P {\exists a \in Atoms\left( x \right)\left( {a \,{\notin}\, y} \right)}$$

19. 19.

Currencies do have a minimal unit for money in circulation. However, other types of monetary scales can have different minimal units. Égré and Barberousse (2014) give the example, from Borel (1907), of wholesale prices as opposed to retail prices. The minimal unit of wholesale prices is smaller than that of retail prices.

20. 20.

Since collective nouns induce discrete scales, it is sufficient to assume a non-standard model extension of $${\mathbb{N}}$$ for my analysis of collective nouns. However, using $${}^{*} {\mathbb{R}}$$ is more convenient because $${\mathbb{N}} \subset{}^{*} {\mathbb{N}} \subset{}^{*} {\mathbb{R}}$$. Furthermore, Fox and Hackl (2006) have argued that our underlying representation of measurement is universally dense, even for discrete measurements. This is consistent with my choice of $${}^{*} {\mathbb{R}}$$ as the underlying model for both cases of qualitative mappings.

21. 21.

For example, it is possible to arrange all the grains of a heap in a row.

22. 22.

That is, if x is an arbitrary collection of grains, but not a heap.

23. 23.

I assume that two individual pluralities of grains of the same size are the same plurality of grains. This raises the question of how to treat different atomic grains. If one assumes only one atomic grain $$g_{1}$$ then $$size\left( {g_{n} } \right) = 1$$ for all $$n \in {\mathbb{N}}$$ because of (i):

(i) $$size\left( {g_{n} } \right) = \left| {Atoms\left( {g_{n} } \right)} \right| = \left| {\left\{ {g_{1} } \right\}} \right| = 1$$

However, this can be addressed by differentiating between the entire individual and its parts. This is due to the fact that every plural individual is the maximal part of itself. Therefore, (41) is well defined, and proper parts of a plural individual can be distinguished by an additional index as in (ii):

(ii) $$g_{n} = g_{1}^{1} \sqcup g_{1}^{2} \sqcup g_{1}^{3} \sqcup \ldots \sqcup g_{1}^{n}$$

24. 24.

$${}^{*} size^{ - 1} \left[ {\mathcal{F}^{c + } } \right]$$ denotes the set of elements of $${\mathcal{F}}^{c+}$$ that are the image of some heap in $$D_e$$ or formally $$\left\{ {H + n \in {\mathcal{F}}^{c + } |{}^{*} size^{ - 1} \left( {H + n} \right) \in D_{e} } \right\}$$, and similarly for $${}^{*}size^{-1}[\mathcal{F}^{+}]$$.

25. 25.

I assume that individuals are of finite size, or formally:

(i) $$\forall x \in D_{e} :\left| {Atoms\left( x \right)} \right| \in {\mathbb{N}}$$An alternative approach along similar lines could assume that a heap of ten thousand grains, and ten thousand grains are different individuals. In this case if $$h_{10{,}000}^{g}$$ is a heap of ten thousand grains then its size would be infinite, as in (ii):

(ii) $$size\left( {h_{10{,}000}^{g} } \right) = H + 10{,}000$$However, such an approach would require a different analysis of plural individuals.

26. 26.

The unspecificity concerns number, though not structure. As mentioned for heap (see above fn. 20), and as a reviewer points out about schools of fish vs packs of wolves, the elements also have a specific structure.

27. 27.

Given that this analysis requires subtracting infinitesimals from real numbers, it is also possible to assume that $$D_{d} ={}^{*} {\mathbb{R}}$$. However, it is not necessary because it is possible to interpret the minus operator ‘$$-$$’ as a function from $${\mathbb{R}} \times{}^{*} {\mathbb{R}}$$ to $${}^{*} {\mathbb{R}}$$ in case that the unit of change is hyperreal.

28. 28.

(63) is a logical consequence of (i):

(i) $${\varvec{tall}} \left( x \right) \ge S\left( {{tall}} \right) \to \forall a \in \left[ {{\varvec{tall}}\left( x \right)} \right]\left( {a \ge S\left( {{tall}} \right)} \right)$$.

29. 29.

Further evidence in favour of the cognitive reality of infinitesimals may be seen in the constant use of infinitely small quantities in the thought of mathematicians throughout the history of mathematics. Lolli (2012) provides a short overview that tracks the use of infinitesimals throughout the history of mathematics: from Archimedes who used the method of exhaustion to compute the area of the parabolic segment; through Cavalieri’s and Galileo’s use of indivisibles; up to Newton’s fluxions and Leibniz’s infinitesimals, which were central in the formation of modern calculus. Infinitesimals fell out of favour, due to a lack of a rigorous definition, in the nineteenth century, until they were given a satisfactory rigorous definition, in 1965, by A. Robinson with the formulation of the hyperreal numbers.

30. 30.

The two heuristic principles that Katz and Sherry (2013) cite are: the law of continuity, which takes one from assignable quantities to non-assignable quantities; and the transcendental law of homogeneity, which takes one from non-assignable quantities to assignable quantities.

31. 31.

The proof is given in (11), by replacing ‘$$\sim$$’ with ‘$$\simeq$$’ and $$I$$ with $${\mathcal{F}}$$.

32. 32.

In a sense, a finite number can be thought of as infinitesimal when compared to an infinite number. However, the extent of the structural similarities between $${}^{*} {{\mathbb{R}} / \mathcal{F}}$$ and $${\mathcal{F}}/I$$ requires further research. One important difference between the two is that $${\mathcal{F}}/I$$ has a unique real number for each equivalence class, but $${\mathbb{R}} \subset \left[ 0 \right]_{ \simeq }$$.

33. 33.

While there is a lot of linguistic and cultural variation in the precise semantic understanding of forest-related vocabulary, an informal survey conducted on the social network indicates that analogues of the tree/grove/forest levels are lexically encoded in Hebrew, Palestinian Arabic, French, Italian, Romanian, Polish, Irish.

34. 34.

In the present case the division between two systems is also connected to the division between two systems of magnitude representation, see Feigenson et al. (2004), and Solt (2016) for more on this.

35. 35.

If $$S,T$$ are rings then $$S \cong T$$ if and only if there is a $$\psi : S \to T$$ s.t that $$\psi$$ is a ring isomorphism. A ring isomorphism is a bijective ring homomorphism.

36. 36.

Let $$S,R$$ be rings then $$\psi : S \to R$$ is a homomorphism if and only if:

(i) $$\psi \left( {a + b} \right) = \psi \left( a \right) + \psi \left( b \right)$$

(ii) $$a \cdot \psi \left( b \right) = \psi \left( {a \cdot b} \right)$$

(iii) $$\psi \left( {1_{S} } \right) = 1_{R}$$

## References

1. Borel, E. (1907). Un paradoxe économique: le sophisme du tas de blé et les vérités statistiques. La Revue du Mois, 4:688–699. English translation by P. Égré and E. Gray, in Erkenntnis (79), 1081–1088, 2014: An economic paradox: the sophism of the heap of wheat and statistical truths.

2. Burnett, H. (2017). Gradability in natural language. Oxford: Oxford University Press.

3. Chierchia, G. (2010). Mass nouns, vagueness and semantic variation. Synthese, 174(1), 99–149.

4. Égré, P. (2015). Vagueness: Why do we believe in tolerance? Journal of Philosophical Logic, 44, 663–679.

5. Égré, P., & Barberousse, A. (2014). Borel on the heap. Erkenntnis, 79(5), 1043–1079.

6. Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314.

7. Fox, D., & Hackl, M. (2006). The universal density of measurement. Linguistics and Philosophy, 29(5), 537–586.

8. Kahneman, D. (2002). Maps of bounded rationality: A perspective on intuitive judgment and choice. Nobel Prize Lecture, 8, 351–401.

9. Kanovei, V., & Shelah, S. (2004). A definable nonstandard model of the reals. The Journal of Symbolic Logic, 69(01), 159–164.

10. Katz, M. G., & Sherry, D. (2013). Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis, 78(3), 571–625.

11. Kennedy, C. (2007). Vagueness and grammar: The semantics of relative and absolute gradable adjectives. Linguistics and Philosophy, 30(1), 1–45.

12. Leibniz, G.W. (1706). Briefwechsel zwischen Leibniz und Des Bosses. In C. I. Gerhardt (Ed.), Die philosophischen Schriften von Gottfried Wilhelm Leibniz, vols. I–VII. Berlin: Weidmann, 1875–1890; reprinted Hildesheim: Georg Olms, 1965.

13. Link, G. (1983). The logical analysis of plurals and mass terms: A lattice-theoretical approach. In R. Bäuerle, C. Schwarze and A. von Stechow (Eds.), Meaning, use, and the interpretation of language (pp. 302–323). Berlin/New York: de Gruyter.

14. Lolli, G. (2012). Infinitesimals and infinites in the history of mathematics: A brief survey. Applied Mathematics and Computation, 218(16), 7979–7988.

15. Magidor, O. (2012). Strict finitism and the happy sorites. Journal of Philosophical Logic, 41(2), 471–491.

16. Nouwen, R. (2014). Plurality. In D. M. Aloni & P. Dekker (Eds.), The Cambridge handbook of formal semantics (pp. 267–284). Cambridge: Cambridge University Press.

17. Pagin, P. (2011). Vagueness and domain restriction. In P. Égré and N. Klinedinst (eds.), Vagueness and Language Use. Palgrave Macmillan (pp. 283–307).

18. Robinson, A. (1974). Non-standard analysis. Princeton: Princeton University Press.

19. Sassoon, G. (2010). Measurement theory in linguistics. Synthese, 174(1), 151–180.

20. Solt, S. (2016). On measurement and quantification: The case of most and more than half. Language, 92(1), 65–100.

21. Sorensen, R. (2012). The sorites and the generic overgeneralization effect. Analysis, 72(3), 444–449.

22. van Rooij, R. (2011). Vagueness and linguistics. In G. Ronzitti (Ed.), Vagueness: A guide (pp. 123–170). Dordrecht: Springer.

23. Williamson, T. (1994). Vagueness. Amsterdam: Blackwell.

## Author information

Authors

### Corresponding author

Correspondence to Yair Itzhaki.

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Yair Itzhaki passed away before the publication of this article. Contact person is Paul Égré (CNRS, ENS, EHESS, PSL University): paul.egre@ens.psl.eu

Note on the paper, by Paul Égré: This paper, written by Yair Itzhaki (1985–2018) between 2015 and 2018, is published posthumously. On January 6, 2018, Yair passed away too soon, after several years of fighting cancer. Yair submitted the manuscript in 2017. Between 2017 and 2018, Yair and I had several meetings to discuss the revisions requested by the journal. Shortly before his untimely death, Yair and his family entrusted me the task of completing the final revisions of his paper. Due to the special circumstances of the publication of this paper, I have performed only minimal revisions to Yair’s original manuscript, with an effort to remain faithful to the points Yair and I had agreed upon in response to the main issues raised by the referees. On behalf of Yair, I would like to thank Naama Friedmann, who made the first few edits of the paper with Yair, as well as Carl Posy, who provided detailed comments, and three anonymous reviewers of the journal for very helpful remarks and criticisms. Many thanks also to Michael Glanzberg and Ede Zimmermann for their editorial assistance, as well as to Nora Boneh, Ivy Sichel and the members of the LLCC group in Jerusalem. Thanks to grants ANR-19-CE28-0019-01 Ambisense, ANR-19-CE28-0004-01 (Probasem), and ANR-17-EURE-0017 FrontCog for support.

Note on the paper, by Yair Itzhaki's family: We express our infinite heap of gratitude to Paul Égré for bringing the paper to light, and for the unique support and mentoring he gave Yair, for believing in him, which were very precious to Yair and brought him joy in his hardest times.

## Appendix

### Appendix

Proof of (9a-d):

1. a.

(i) Property: $$\forall\,\, a,b \in {{\mathcal{F}}}:a \pm b \in {\mathcal{F}}$$

Proof: $$a,b \in {\mathcal{F}} \Rightarrow \exists N_{1} ,N_{2} \in {\mathbb{N}}:\left| a \right| < N_{1} \wedge \left| b \right| < N_{2} \Rightarrow \left| {a \pm b} \right| \le \left| a \right| + \left| b \right| < N_{1} + N_{2} \Rightarrow a \pm b \in {\mathcal{F}}.$$

(ii) Property: $$\forall a,b \in {\mathcal{F}}:a \cdot b \in {{\mathcal{F}}}$$

Proof: $$a,b \in {\mathcal{F}} \Rightarrow \exists N_{1} ,N_{2} \in {\mathbb{N}}:\left| a \right| < N_{1} \wedge \left| b \right| < N_{2} \Rightarrow \left| {a\cdot b} \right| = \left| a \right| \cdot \left| b \right| < N_{1}\cdot N_{2} \Rightarrow a\cdot b \in {\mathcal{F}}.$$

2. b.

(i) Property: $$\forall \epsilon ,\delta \in I:\epsilon \pm \delta \in I$$

Proof by contradiction: Let $$\epsilon ,\delta \in I$$ be s.t. $$\epsilon \pm \delta \,{\notin}\, I$$, it follows that

$$\exists N \in {\mathbb{N}}:\frac{1}{N} < \hbox{min} \left\{ {\left| {\epsilon + \delta } \right|,\left| {\epsilon - \delta } \right|} \right\}$$, but $$\left| \epsilon \right|,\left| \delta \right| < \frac{1}{4N} \Rightarrow \left| {\epsilon \pm \delta } \right| \le \left| \epsilon \right| + \left| \delta \right| < \frac{1}{2N} < \frac{1}{N} .$$

(ii) Property: $$\forall \epsilon ,\delta \in I:\epsilon \cdot \delta \in I$$

Proof by contradiction: Let $$\epsilon ,\delta \in I$$ be s.t. $$\epsilon \cdot \delta \,{\notin}\, I$$, it follows that

$$\exists N \in {\mathbb{N}}:\frac{1}{N} < \left| {\epsilon \cdot \delta } \right|$$ but $$\left| \epsilon \right|,\left| \delta \right| < \frac{1}{N} \Rightarrow \left| {\epsilon \cdot \delta } \right| < \frac{1}{{N^{2} }} < \frac{1}{N}.$$

3. c.

Property:$$\forall \epsilon \in I:\forall a \in {{\mathcal{F}}}:\epsilon \cdot a \in I$$

Proof by contradiction: Let $$\epsilon \in I,a \in {\mathcal{F}}$$ be s.t. $$\epsilon \cdot a \,{\notin}\, I$$, it follows that $$\exists N_{1} \in {\mathbb{N}}:\frac{1}{{N_{1} }} < \left| {\epsilon \cdot a} \right|$$, but $$\exists N_{2} \in {\mathbb{N}}:\left| a \right| < N_{2} ,\left| \epsilon \right| < \frac{1}{{2N_{1} N_{2} }} \Rightarrow \left| {a\cdot\epsilon } \right| < \frac{{N_{2} }}{{2N_{1} N_{2} }} = \frac{1}{{2N_{1} }} < \frac{1}{{N_{1} }}.$$

4. d.

(i) Property: $$\forall H \in {{\mathcal{F}}}^{c} :\forall a \in F:H \pm a \in {{\mathcal{F}}}^{c}$$

Proof by contradiction: Let $$H \in {\mathcal{F}}^{c} ,a \in {\mathcal{F}}$$ be s.t $$H \pm a \in {{\mathcal{F}}}$$, it follows that $$\exists N_{1} \in {\mathbb{N}}:\hbox{max} \left\{ {\left| {H + a} \right|,\left| {H - a} \right|} \right\} < N_{1}$$, but $$\left| H \right| \le \hbox{max} \left\{ {\left| {H + a} \right|,\left| {H - a} \right|} \right\} < N_{1} \Rightarrow H \in {{\mathcal{F}}} .$$

(ii) Property: $$\forall H \in {{\mathcal{F}}}^{c} :\forall a \ne 0 \in F:H \cdot a \in {{\mathcal{F}}}^{c}$$

Proof by contradiction: Let $$H \in {\mathcal{F}}^{c} ,a \ne 0 \in {\mathcal{F}}$$ be s.t $$H \cdot a \in {{\mathcal{F}}}$$, it follows that

$$\exists N_{1} \in {\mathbb{N}}:\left| {Ha} \right| < N_{1} \Rightarrow \left| H \right| < \frac{{N_{1} }}{\left| a \right|} \Rightarrow H \in {{\mathcal{F}}}.$$

Proof of (13):

Lemma: $$\forall a \in {{\mathcal{F}}}:\exists r \in {\mathbb{R}}:r \in \left[ a \right]$$

1. (i)

Let $$a \in {\mathcal{F}}$$ be s.t.$$\forall r \in {\mathbb{R}}:r \,{\notin}\, \left[ a \right]$$.

2. (*)

Without loss of generality let $$a$$ be s.t. $$0 \le a$$, otherwise substitute $$a$$ with $$- a$$

3. (ii)

$$a \in {\mathcal{F}} \Rightarrow \exists N \in {\mathbb{N}}:a < N$$ (from the definition of $${\mathcal{F}}$$,$$\left( * \right)$$)

4. (iii)

$$N \in {\mathbb{R}} \Rightarrow N \,{\notin}\, \left[ a \right]$$ (from (i) and $${\mathbb{N}} \subset {\mathbb{R}}$$)

5. (iv)

$$\forall b \in \left[ a \right]:b < N$$ (from (ii), (iii), (9a))

6. (v)

Let $$A$$ be the set of the real upper bounds of $$\left[ a \right]$$, that is $$A = \left\{ {s \in {\mathbb{R}} |\forall b \in \left[ a \right]:b \le s} \right\}$$

7. (vi)

$$N \in A, {\text{so }}A \ne \emptyset$$ (from (iv))

8. (vii)

$$\exists R \in {\mathbb{R}} \cap A:R = \sup \left( {\left[ a \right]} \right)$$ (from (vi), the fact that [a] is nonempty, and the least upper bound property of $${\mathbb{R}}$$)

9. (viii)

$$R - a \,{\notin}\, I$$ (from $$\left( {\text{i}} \right)$$)

10. (ix)

$$\exists M \in {\mathbb{N}}:\frac{1}{M} < R - a$$ (from (viii),$$\left( * \right)$$)

11. (x)

$$a < R - \frac{1}{M} < R$$ (from (ix))

12. (xi)

$$R - \frac{1}{M} \,{\notin}\, \left[ a \right]$$ (from (i), $$R \in {\mathbb{R}},M \in {\mathbb{N}}$$)

13. (xii)

$$R - \frac{1}{M} \in A$$(from (x), (xi))

14. (xiv)

$$\bot$$ (from (vii), (x), (xii)).

Proof that $${\mathbb{R}} \cong {\mathcal{F}}/I$$:Footnote 35

Lemma I: $$\forall a,b \in {\mathcal{F}}:\left[ {a + b} \right] = \left[ a \right] + \left[ b \right]$$

Proof:

1. (i)

$$\left[ {a + b} \right] \subseteq \left[ a \right] + \left[ b \right]$$:

$$c \in \left[ {a + b} \right] \Rightarrow \exists \epsilon \in I:$$ $$c = \left( {a + b} \right) + \epsilon = \left( {a + \frac{1}{2\epsilon }} \right) + \left( {b + \frac{1}{2\epsilon }} \right) \in \left[ a \right] + \left[ b \right] .$$

2. (ii)

$$\left[ a \right] + \left[ b \right] \subseteq \left[ {a + b} \right]:$$

$$c \in \left[ a \right] + \left[ b \right] \Rightarrow \exists \epsilon ,\delta \in I$$: $$c = \left( {a + \epsilon } \right) + \left( {b + \delta } \right) = \left( {a + b} \right) + \left( {\epsilon + \delta } \right) \in \left[ {a + b} \right].$$

Lemma II: $$\forall a,b \in {\mathcal{F}}:a \cdot \left[ b \right] = \left[ {a \cdot b} \right]$$

Proof:

1. (i)

$$a \cdot \left[ b \right] \subseteq \left[ {a \cdot b} \right]$$:

$$c \in a \cdot \left[ b \right] \Rightarrow \exists \epsilon \in I$$: $$c = a\cdot \left( {b + \epsilon} \right) = \left( {a\cdot b} \right) + \left( {a\cdot \epsilon} \right) \in \left[ {a \cdot b} \right].$$

2. (ii)

$$\left[ {a \cdot b} \right] \subseteq a \cdot \left[ b \right]$$:

$$if\,\, a = 0$$:

$$c \in \left[ {0 \cdot b} \right] = \left[ 0 \right] \Rightarrow c = 0\cdot \left( {b + 0} \right) \in 0 \cdot \left[ b \right].$$

$$\hbox{if}\,\, a \ne 0$$:

$$c \in \left[ {a \cdot b} \right] \Rightarrow \exists \epsilon \in I$$: $$c = a\cdot b + \epsilon = a\cdot \left( {b + \frac{\epsilon} {a}} \right) \in a \cdot \left[ b \right]$$.

Let $$\psi : {\mathbb{R}} \to \mathcal{F}/I$$ be s.t. $$\psi \left( r \right) = \left[ r \right]$$

Proof that $$\psi$$ is a ring homomorphism: Footnote 36

1. (i)

$$\psi \left( {r + s} \right) = \psi \left( r \right) + \psi \left( s \right)$$:

$$\psi \left( {r + s} \right) = \left[ {r + s} \right] = \left[ r \right] + \left[ s \right] = \psi \left( r \right) + \psi \left( s \right).$$

2. (ii)

$$r \cdot \psi \left( s \right) = \psi \left( {r\cdot s} \right):$$

$$r \cdot \psi \left( s \right) = r \cdot \left[ s \right] = \left[ {r\cdot s} \right] = \psi \left( {r\cdot s} \right) .$$

3. (iii)

$$\psi \left( 1 \right) = \left[ 1 \right] \left( {\text{from}}\,{\text{the}}\,{\text{definition}}\,{\text{of}}\,\psi \right)$$

Therefore, $$\psi$$ is a homomorphism.

Proof that $$\psi$$ is bijective:

1. (i)

$$\psi$$ is injective:

$$\psi \left( r \right) = \psi \left( s \right) \Rightarrow \left[ r \right] = \left[ s \right]\mathop \Rightarrow \limits_{{\left( {14} \right)}} r = s .$$

2. (ii)

$$\psi$$ is surjective:

$$\forall \left[ r \right] \in {\mathcal{F}}/I:\psi \left( r \right) = \left[ r \right].$$

Therefore, $$\psi$$ is an isomorphism.

Therefore, $${\mathbb{R}} \cong {\mathcal{F}}/I.$$

## Rights and permissions

Reprints and Permissions